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K.BIBI. . RADCL.
iBODU Liafi.
1 DUPUC*:rE
1 80LD BY I
CHARLES H.Mtf KIM
OXFORD
MUSEUM.
LIBRARY AND
READING-ROOM.
JHIS Book belongs to the "Student's
Library."
1 It may not be
removed from the
Reading Room
without permission
. of the Librarian.
-
/-
COLLECTION op EXAMPLES
or
THE APPLICATIONS
OF THE
CALCULUS OP FINITE DIFFERENCES.
BY
J. P. W. HERSCHEL, a.m. f.r.s. l.&e. f.c.p.s.
REG. SOC. GOTTING. CORRE8P. SECRETARY OF THE ASTRONOMICAL SOCIETY,
AND FELLOW OP ST. JOHN'S COLLEGE, CAMBRIDGE.
CAMBRIDGE:
Printed by J. Smith, Printer lo the University ;
AND SOLD BY J.DEIGHTON & SONS, CAMBRIDGE; G. & W. B. WHITTAKEK,
^VE.MARfA LANE; J. MAWMAN, LUDGATE STREET; AND LONGMAN 1^ CO.
' PATERNOSTER ROW, LONDON.
J8'20
P R E F A C E
In applying the general principles of the Calculus of
^Differences, laid down in the Appendix to the Translation
of Lacroix, we have assigned the first place to Examples
purely analytical of the methods themselves^ endeavouring
always to select such as may not only be useful to the stu-
dent as exercises, but also as results, which he may have
occasion to refer to in his future enquiries, and will thus be
valuable in themselves, as materials which he will find it
advantageous in a more advanced state of his knowledge to
have ready at hand. Many of these results are theorems of
some generality which we believe will not be found else-
where, or at least, are not in common use, and where these
occur, the leading steps of the demonstration are either set
down, or the principle on which it depends mentioned, and
in both cases the student will find a useful exercise for his
invention in supplying what is omitted. We have then pro-
ceeded to questions of a more mixed nature, illustrative of
the application of the Calculus of Differences, to a variety of
subjects in which it may be employed with advantage as an
instrument of investigation, such as the determination of the
general terms of the series when the law of their formation
is given, the theory of circulating functions, continued frac-
tions, the determination of curves from such properties as
involve a series of consecutive points separated by finite
# ^i
iv ' piitFAct:.
intervals^ the doctrine of Interest and Annuities^ and such
other subjects as can be properly treated within our limits.
The want of a rejgular treatise, on the Calculus of Differ-
ences in our language^ has long been a serious obstacle to
the progress of the enquiring student. The Appendix
annexed to the translation of Lacroix's Differential and
Integral Calculus, although from the necessity of studying
compression it is not so complete as its author could have
mshed,^ will, it is hoped, remove this obstacle in somedegree^
and at least put the analytical principles of the pure theoretical
part of the calculus in the reader's power. But the method
of applying those principles^ and the formulae derived from
them^ to the various cases and questions of pure, as well
as mixed mathematics^ in which they may be advantageously
introduced, also demands some degree of explanation, and
accordingly, in such of our examples as have this for their
object, the reader will find the successive steps of the pro-
cesses fully detailed, till the questions are reduced to such
analytical difficulties, as the Appendix, or the preceding
problems will enable him to surmount by himself.
CONTENTS.
PART III.
1. On the direct method of Differences . . . . ^ ^ ^ • z
S» On the Resolution of. Functions into Factorials to prepare
them for Integration -. •••... 13
3 k On the Reduction of Fractional Expressions to Integrable
Forms » 4 ^g
4* On the Integration' of Equations of Differences". . i . • 31
<5. On the Integration of Equations of mixed Differences . ^ • Z7
6« On the Summation of Series by the Integration of theii"
general terms k ^ . . « 43
7* Problems and Theorems relating to the developemeat of
Exponential Functions* and the properties of the num-
bers comprised in the form A*" 0" . . ... * . ; . 66
8* Application of the Theorems in Sect. 7, to the developement
of Particular Functions* the Summation of Series* &c. * • 79
*
9. On the Interpolation of Series « 99
1 0. Application of the Calculus of Differences to the determina*
tion of Curves from properties involving consecutive points
separated by a finite interval 107
11. On Circulating Equations 137
12. On continued Fractions 148
13. Application of the Calculus of Differences to various
Problems 156
PART III.
SECTION I.
EXAMPLES OF THE DIRECT METHOD OF DIFFERENCES.
In the following questions, x is supposed to be the
independent variable, so that A x is unity throughout.
(1). Required the differences of sin •rd and cos x0,
^ being constant,
A sin X 6s=2 . sin :: . cos
2
(-+1)
A cosx^= — 2 . sin- sin (^^+5).
(2). Required the differences of sin (h -{- x $) and
cos (h+x$)
Asin(A+r^) = 2 .sin - .cos<A+(j:+ -V |
Acos(A+jea)=-2 .sin -.sin|A+rj^+ 5)^}.
(3). To find the (« h)'^ and (2 «- 1^^ differences of the
same functions, . . -- z ^.. . •
'A*» sin {h -h x0) - ^2 sin V\ . sin { A + (a: + «) ^ {
2 sin 3^ .cos {A+(«r + 11)^1
A**- 'sm(A + T^)=:
* A
2
(2 sin -J . cos
2.
{*.(.. i^.)}
A»— » COS (A + «r a)
= — ^2 sin -J . sin < A -f r «
2«-
i.)}.
In like manner we maj obtain the following diiFerences:
(4). A tan x0 =
sin
cos xd . cos (x •\- I) e*
(5). A cotan xd :=: -:
— sinO
(6)- A
a* 4-1
« — 1
sin*t . 8in(ai'+I)tf
2o<
«'' + ■-!
(7). a!>L±1>=2'.2:J-L.
«»'-! a*'+ 1
(8). A.«'rini.=«' + «.8m^.(.in^)'.
(9). ^.2»'0in^y-2- + «.(sin^)V
(JO). ^ttni«
tan
£* + »
ii > >
(11). A.tan-;«-a'.taii-.(tan-^)
(12). A. (-«)•. sin 1
(-iy+'.2'+» .8in54r;(«>»2^.y.
3
e
sin — 2-,
(19). A i =-2 ?— -
B d
2* . sin — 2' . sin —
tan
1 p' + >
(H). A — i — 3 = *
2'.tan-^ * ^
. .
COS -- sin
(16). A -J «1_ = «Ltl__.
(17.) Acot.'. = -_L_.
V •■' «» - «• + • •
(19.) A '
(20.) Let /,s sin (a -f X 0) ; r,ss cos (a +x 6), then
1
^# • ^# 4- 1 • • • • ^« ^ M ^
I
2ttn(i»+i)^
• ^' ' ■■ ■ III ■ ill ■
O • ^* 4- 1 • • • ' ^* + 1» + i
(filO The same notation being employed,
A-i=i>l=2,co,g.^-»)'-^'
(22). A — Ljy —
/, . J*, 4. 1 . . . . /jr ^ fl„ _|. 1
= 2 . co$ (n.+ 1)0. (-')"*•' -J^+. + i
•^jr • "^jr + 1 • • • • ''^^ + a» +
(2S). A -^^ZliL = 2 . cos . ^— ^^"^ '
(34). A
^a • ^* + 1 • • • • ^Jr + 9» + 1
= 2.CO8(« + l)0.-i i '''' + ^ + ^ ,
(25). Required the diflFerence of arc tan x 6, or (as wc
shall in future designate this function) tan"*^ x 6.
Atan-'jrg=tan~' -^ ^^ - ^ A .
The expression tan "^ x 6 must not be confounded with
(tan x6)--^ or . The reason^ for employing this mode
tan X d
of expression instead of the geometrical circumlocution arc
(tan=:j: 6) or arc tan x 6 are these.
We have already in the differential calculus as well as
in that of differences, experienced the great advantage not
only in point of brevity, but of clearness and symmetry which
arises from denoting the repetition of the operations expressed
by d and A, by annexing the number of repetitions as an
exponent to the characteristic, and we have already seen
(Appendix, Art 3780 that the inverse operation of integra-
tion in the two calculi is rightly represented on this principle
by the same characteristics d and A with negative exponents.
The same notation may be used to denote the repetition of
any operation, whether it be such as modifies the /arm of a
function, which is the case with those just noticed or such as
expresses the nature of the function itself, as log, cos, tan>
r. ■»
&c. We may use -log* jr, cos' x, tan** x, &c. for log log x,
cos cos cos X, tan tan tan x, &c. respectively, and in
general
/(/ (^)) o' ffi^) ^^7 ^^ written/" (x)
/ (/ ( / (^))) or fff (x) may be written / ' (*)
and so on, which gives in general/*/* (a:) =/**•♦■*(*).
If we now enquire, the meaning of /* (a:), we need only
make iy=0, m = l, which gives
//W=/(^), .
and consequently /*» (x) = x. If now we make w = 1 and
n=- 1 we find
//-«(x)=/>(x) = x,
so that/'"'"(x) must denote ** that quantity whose function /
is x/* or rather, that function of x which operated on in the
manner denoted by/" shall produce simply j?. /*-"* (x) then
denotes the inverse function of /"(x): thus tan""*x will stand
for arc (tan=x), sin*"^ x, cos ""* x, log""^ a? respectively for
arc (sin =s x), arc (cos = x), ^, or number whose logarithm
is X *. The symmetry of this notation and above all the new
* The notations /* (j?), /"•* (x), sin"! jr &c. were explained
by the author of these latter pages in a paper, '' On a remarkable
apptication of Cotes^s theorem'' in the Philosophical Transactions,
1813, as he then supposed for the first time. The work of
a German Analyst, Burmann, has, however, within these few
months come to his knowledge, in which the same is explained
at a considerably earlier date. He, however, does not seem to
have noticed the convenience of applying this idea to the inverse
functions tank's &c. nor does he appear at all aware of the inverse
calculus of functions to which it give rise. Burmann is a zealous
partizan of the combinatory analysis of Hindenburg, tlie very
principle of which is the exclusion of all analytical artifice^ which
is no where so strongly called for as in the calculus here alluded to*
stnd most extensive yiews it opens of the nature of analytical
operations seem to authorize its universal adoption^ not to
mention the real inconvenience which more than one author
of eminence has been put to for want of some notation
founded on principle to express any inverse function without
introducing a new characteristic.
4 ft
The equation which gare rise to this digression is easily
proved if we call to mind that
-. /A Ti\ tan -rf- tan 5
tan (il - li) = ^ ;
1 + tan ^ . tan JB
for, if we take the inverse function tan "" * on both sides, we
liave
An 1 ^ tan -rf— tan B 1
-4-JB s tan-*< -, r,l
C. I + tan u4 . tan i5 3
and for A and B writing tan""* A and tan"^ B^
tan-* -rf-tan-* J3=tan-*^ —r^i
(.1 + JBy
which the student is left to apply to the case in question.
(26). Required the difference of tan"- * (A + x 0).
A tan- * (h + x ^) = tan- » j —rTm m Tj :^ C .
V
(27.) Required the difference of tan — * < -^ — ^ > ,
+
Ab-aB
tan""
(28). Required the difference of tan "" * .«,
A tan-* a, = tan-'-— f .
l+t«*«^ + i
(29>. Required the difference of 2' .tan - * f ^ j
A . «' tan- (I) , 2- . tan- {^-^,-^5^^.} .
(30). It is required to demonstrate the truth of the two
following theorems in which j9 represents 2 sin ^»
cos 2 «^ c= I ^ np,sine — l^^LzD j,^ . cos 2
+ — 1 LI L p9 . Sin 3 ^ + &c.
sin 2 « = - p , cos a - ^L^^Hilp" . sin 2 ^
the sign being alternately +4- and the series breaking
off at f^ whenever » is a positive integer.
These may be deduced from the general expression for
<<«4-i» in Art. 345) by substituting cos 9. x6 or sin 2x6 for
u„ developing their successive differences and finally making
x==0.
(31). Required the successive diflFerences of x*, a^^ x*,
jf^j expressed in powers of ^, and the law of their coefBcientr.
i
A. x»cs8a?' + 3j?+l. A*, dj'zre a: + 6, A'.j:»=6,
A. x*=4r»4-6 x* + 4i:-fli A*.^*=:12 ap* + 24 jr + 14,
A*. «*=24.r+36. A* . «*= 24, \
A».««=:A«(?'» + ^A»(?^'-'.^+^!i^^IlilA» (>•«-*. «»+...
1 1 .2
^ m{m-\) («-^O A«ffH ti^-».
1.2... .(w — «)
and most extenuve views it opens of the nature of analytical
operation* seem to authorize its universal adoption, not to
mention the real inconvenience viiich more than one author
of eminence has been put to for want of some notation
founded on principle to express any inverse function without
introducing a new characteristic.
The equation which gave rise to this digression is eatilf
proved if we call to mind that
1 -I- tan ^ . tan -B
for, if we take the inverse function tan ~ ' on both sides, we
liave
■A » . -. ( tan-^-tan B 7
A—B = tan — ' -J , j^ f
tl -(- tan ^ . tan B >
and for A and B writing tan ~ ' 4 and tan ~ ' B,
which the student is left to apply to the case in question.
(26). Required the difference of Un — ' (A + 1 6).
^'"°"''*+"> = '""".'{r+57*+iWM^?-
(27.) Required the diflference of tan — ' '< ^ — ,5- > .
A tan — '■?-- jr- > a:
Lji + BxS
(SB)- Required the difference of Ian ~
Au
& tan ~ ' u, = tan
(29). Required the difference of 2', tan-' T^)
i . r «„-. (1) = r . un- { ,,,„;3^., -} .
(30). It is required to demonstrate the truth of the two
following dieorems in which jp represents 2 sin 9,
co8a«fl = l - tip.iinO — " ■ p' . cos a fl
B(«-l)(H-g)
p= . sin 3 e + 8w-
1,8.9
sin 9 « 9 = - p . cos fl ~ "^"~?..p' . wn 2 fl
_ "t"-iM»--y . cos 3 + 8ic.
1 .2.3 '^
the sign bdng alternately + + and the series breaking
off* at p" whenever « is a positive integer.
These may be deduced from the general expression fw
u,^^in Att.SiS, by substituting cos 2x0or sin 2xBfor
u,, developing their successive differences and finally making
1=0.
{31). Requited the successive differences of «*, a*, x*,
x", expressed in powers of x, and the law of thetr coefficientr.
A. x'mS*» + 3«+l, A'.ii»=6 a: + 6, A».ir'=6,
' A.i*=4a;'+6a?» + *x + l, A*. J^>=12 *'+24 «+I4,
A* . ««=24 *+ 36, A* . «*= 24, >
^^ BT.jTa
&M-*A-«-+ ^A-.— .X+ ^^:!Lli^A-tf— >.
this last equation may be derived from the value of A" . x*"
given in the Appendix, Art. 350, by developing its several
terms by the binomial theorem, and collecting the coefficients
of similar powers of x. If we then call to mind the defini-
tion given in that article of the expression A" o*^ the truth of
the above equation will be apparent. But the most regular
as well as the easiest mode of obtaining it, is from the gene-
ral theorem of Arti 36 1.
(32). To prave the truth of the following theorems
(sin Oy . cos ndzn
(sin 6y . sin « 6 =
ll....(/r+l) l....(/i + 3) >
these may also be deduced from the same general expression
for A^u, by substituting for Uj^, sin ar 6 and cos <r By and re-
placing A^ti, by the value given in Example (3).
The numbers comprised in the form A* o"^ enter so ex-
tensively into the theory of series, and afford such remarkable
facilities in the developement of functions that we shall: take
occasion to annex in this place a short table of their values
as far as A*®^^® and hereafter to present the most remarkable
of their properties, sufficient to explain the manner of their
employment with reference to this object.
(33). To calculate the actual numerical values of A" cT
for all values bf m and n from I up to 10.
'<
i
<J
i
1
<
1
1
1
<1
1
1
i
<1
1
g
1
1
1
<1
s
1
1
1
i
5
i
<
■#
s
1
g
s
?
S
i
s
2
^
^
g
3
%
1
1
i
lO
^
«
«=
Tl-
S
s
a
s
o
1
<
-
-
- -
-
-
-
-
-
-
-'
fc
o 5
a.
•o
4.
■o
&
b
10
The inspection of this table affords room for one or two
remarks. The value of the function A'^o"' increases with
the indices n and m, but is more influenced by the latter than
the former* The expressions A*o*, A*o* + ", A*«* + %&c.
go on perpetually increasing in a very high ratio, and end in
surpassing any assignable number. Their rate of increase
too surpasses at last that of any assignable geometric pro-
gression, as we shall soon see.
(34). To assign the approximate values of A" o% A* o* + ^
AV + % &c. when n is a very high number.
This may easily be done, if we actually develope (/— 1)%
or
I Vl . 2 1.2.8 />
by the binomial theorem, and compare the coefficients of the
powers of / so produced with those of the same powers in
the series,
''\r:Tl.-' i....inW ''''h Art.S61.App.
for we thus obtain
A*o*=s 1,2 n,
A*c»» + '=s 1.2....(a+l) X-,
A»fl^ + «= 1 .2....(i» + 2) X ?"*•*•".
24 *
A» «?• + »« 1 ...... (« + S) X 2l±>l\
48 '
5760 *
11
As n increases, these functions therefore increase ulti-
mately at the same rate with those progressions spoken of in
the Appendix under the name of hyper-geometrical series
(Art. 414.) and when n is very large^ we get, by applying the
Formulae of Art. 411.
2 \ e ^
8 V ^ /
wr, if we consider that when « is a very high number,
(n +!)»=: 6. n»,(/f + 4)»=V.»\ &c.
48 Key
If more exact formulae be requiredi the series of powers
of - must be taken into the account.
(55). To shew that
1 1 • «
.1 • % • u>
n
The generating function of Uj, being <p (<), that of u,^»is
— (/). Let this be thrown into the form
and developed by the binomial theorem. If we then re-
descend from the generating functions to the coefficients of
f in their developements, the theorem in question will result.
{36), To prove in general that
M. i • ^
.(« + 3>-l)(» + 3r-^)^3
1.2.3 ^^
The generating function - ^ (i) of w^ + „ must be trans-
V
formed into a series of terms of the form
and their coefficients shewn to coincide with those of the
proposed series. To develope /""'* in powers of v( 1 )
put the latter function equal to z and we have to develope
- ) in powers of z, - being a function of z given by the
equation
'■G-')==
.,i=,..-.(iy.
13
Lagrange's theorem demonstrated in Note E enables us to do
this. If we put y for - , we have
which gives
2/«= 1= 1 + ?:. + ^U4-2r-l)
^ «" 1 1.2
whence the theorem in question results.
SECTION II.
Exercises in the resolution of Inunctions into Fac-^
torials, to prepare them for Integration.
(1). To resolve j?% j:% j^, j?% into products of the factors
x^ J7— 1> x—% &c., or as it is sometimes expressed^ to reduce
them to 9c and its preceding values ^
x» = x + 3 X (r— \)^-x (j - 1) (x-2)
x*=x + 7'»(x-l) + 6x(x-l)(«-2)+x(x--l)(;v-2)(x-S)
Ao* A*«* V A'c*
x*=^ r + — ^z(xT-l) +-i-^^(*-l)C^-2) + &c.
The general expression may be deduced from the equa-
tion
(See Appendix, Art. 345), by making «^=i/' and then sup-
12
The generating function of u^ l)eing <p {t), that of u«+«i8
— <p (/). Let this be thrown into the form
Ji-/(2-i)]".^(0
and developed by the binomial theorem. If we then re-
descend from the generating functions to the coefficients of
f in their developements, the theorem in question will result.
{36), To prove in general that
.(« + 3.-l)(» + 3_r-^)^3
1.2.3 ^
The generating function - (0 of w, + « must be trans-
V
formed into a series of terms of the form
and their coefficients shewn to coincide with those of the
proposed series. To develope /"~" in powers of v( 1 1
put the latter function equal to z and we have to develope
- ) in powers of z, - being a function of i given by the
equation
or
13
Lagrange's theorem demonstrated in Note E enables us to do
this. If we put y for - , we have
which gives
^ «" 1 1.2
whence the theorem in question results.
SECTION II.
Exercises in the resolution of Functions into Fac-^
torials, to prepare them for Integration.
(1). To resolve a?% j:', j?*, j?", into products of the factors
x^ J7— 1> x—2y &C.9 or as it is sometimes expressed^ to reduce
them to X and its preceding values ^
x*=:x+J? (*— 1)
x» = x + 3 X (r— \)^-x (x - 1) (x — 2)
x*=x + 7'»(j:-l) + 6x(d?-l)(«— 2)+x(x— l)(Ar— 2)(x-S)
A /J* A* />* A' /J*
The general expression may be deduced from the equa-
tion
«, + ^=«/+ 7 ^«, + ^-^^ A*w,+ &c.
(See Appendix^ Art. 345), by making 1/^=^" and then sup-
14
t>08ing y=0. The values of A^", A*o% Sec. when n does
not exceed 10 may be taken from the table given in page (9).
(2), To resolve x, x\ x', x*^ a:* into products of the suc-
cessive factors x+1, •x' + 2, &c. or to reduce them to /«r-
ceeding values of x.
**= + 1 -3 (;? + 1) +(j?+ 1) (r + 2)
«*= - 1 +7 (J? + l)--6 (x+l)(jr + 2) + {x+ 1) (x+2) (*+S)
• = + 1 — 15(^+1) + 25 (*+l)(a:+2)--10(x+l)..(j? + 3) +
4-(^ + l)...-(*+4)
«»«t(— 1)** J> ; (ir + 1)
^ i 1 1 .2
A* (?* + ' ")
This may be deduced from the same general expression '
for tty + , by putting — .r f or j: and supposing m^ = y* + » and
then proceeding as above.
(3). It has been remarked (App. Art. 370.) that *' to
keep the numerical coefficients as low as possible in these
reductions to the form of factorials is an object of import-
ance," and that " this may be done by a proper disposition of
the preceding and succeeding factors." The following ex-
amples will shew how this is to be performed.
cr»=:(jr-2)(a:-l)x(x + l)(a?+ 2) + 5 (j:-- 1) ar(j:+l) + jr.
«'=s(x-S)...(r + S)+ 14(a:-2)...(jr + 2) +
In these instances it will be observed that no factorials .
with an even number of factors enter into the expression.
15
This is a simplification of considerable moment, and that it
takes place in general for x'* + * may be demonstated with-
out difficulty as follows.
•
(4). To resolve ar** + * in the same manner, and to
determine the law of the coefficients.
Assume j?^ = V. Then af^* "*" ' = v"* x^ suppose now
v" = J.-k'A, (v-l) + u^, (v-i) (t;-4) +
+ A^ (v-l) (v-4) (v-9) + ^»(v-l)(i;-4). .(«— n*)
This assumption is possible because the second member is
a rational integral function of v of the 72th degree, and being
reduced to powers of v, and compared with v" will afford
n + 1 equations of the first degree for the determination of
the indeterminate coefficients A^^ A^^ .... A^. The follow*
ing is however a readier and inore elegant method. The
above equation being identical in v must hold good what-
ever numbers are substituted for v, hence if for v we write
in succession 1, 4, 9, 25, &c.j we get
3*«=:^, + (3*-.l») A, + (S*- ]•) (S*--2«) A^
&c. = &c.
Whence we derive
-4. = i", .
0«M 1«»
A,^— + _i ,
2« - 1» l»— 2« '
t«» oi»
1
and so on, the law being evident and the general value of A,
susceptible of direct expression in functions pf or.
X'l
14
^ding i/asO. The values of A 9% A*o% &c. when n does
not exceed 10 may be taken from the table given in page (9).
(2). To resolve x, ar% x*, x*, a:* into products of the suc-
cessive factors j:+ 1, * + 2, &c. or to reduce them to suc^
ceeding values of x.
«•= — l+(^ + l)
jf » =: + I ~ 3 (;v + 1 ) + (« + 1 ) ( r + 2)
j:*=: - 1 +7 (x + l)-6 (x+ l)(ar+ 2) + (x+ 1) (x+2) (* + S)
• = + 1 — 15 (J?+l) + 25 (a?+l) (x+2)— 10 (x+1). . (j: + S) +
+ (x + l) (x+4)
A* rf* + • ■)
+ -VT (* + ^) (^ +'«)-&c. {
This may be deduced from the same general expression '
for My + , by putting — .a? for x and supposing m^ = t/* + * and
then proceeding as above.
(3), It has been remarked (App. Art. 370.) that *' to
keep the numerical coefficients as low as possible in these
reductions to the form of factorials is an object of import-
ance," and that ^^ this may be done by a proper disposition of
the preceding and succeeding factors/^ The following ex-
amples will shew how this is to be performed.
jr»=(x— l)x(x + i) + x.
;r*5=(x-2)(a:-l)x(x + l)(4?+ 2) + 5 (x-. 1) *(x+l) + jr.
«' = (jr- 3). . . (x + 3) + 14 (X - 2). . . (JT + 2) +
+ 2I(«— l)x(jr + l) +«.
In these instances it will be observed that no factorials .
with an even number of factors enter into the expression.
15
This is a simplification of considerable moment, and that it
takes place in general for x'* + * may be demonstated with-
out difficulty as follows.
(4). To resolve ar** + * in the same manner, and to
determine the law of the coefficients.
Assume z^ssv. Then af** "*" ' = v** x, suppose now
v* =:J^ + ji, (V- 1) + J^ (V- 1) (v-4) +
+ J^ (v-1) (u-4) (v-9) + ^»(v-l)(i;-4). .(t>— n*)
This assumption is possible because the second member is
a rational integral function of v of the 72th degree, and being
reduced to powers of v, and compared with v" will afford
n + 1 equations of the -first degree for the determination of
the indeterminate coefficients A^^ -4„. . ..A^. The follow*
ing is however a readier and more elegant method. The
above equation being identical in v must hold good what-
ever numbers are substituted for v, hence if for v we write
in succession 1, 4, 9, 25, &c.j we get
3*«=^, + (3*-.l») A. + iS^-V) (S*--2«) A^
&c. = &c.
Whence we derive
i«M 1«»
own !•»
A,^— + _i ,
2« — 1» l»— 2« '
l«» o«»
1
(3*— l«)(S*-2«) (a*- l*)(2«-3») (!•-«•) (1«-S«)'
and 80 on, the law being evident and the general value of A,
susceptible of direct expression in functions of x.
16
Such then being the values of A^, A^^ &C.9 we have
=: J^x + A, (x~l)x(jr + l) + &c.
V being equal to x*.
The law observed by the coefficients is not a little re-
markable. It extends too with a slight modification to cases
of much greater generality, and it will hardly be thought
irrelevant to the present subject to propose and resolve the
following problem.
(5). To develope F (x) in a series of factorial terms of
the following form
F{x) =^Ao + A (*-/) + J, (X -/) (X -/.) + &c.
•
F(x) being any function whatever of x, and /„ y^, &c. par- ^
ticular values of any other function f, corresponding to the
values 1,2, See. of x,
A process exactly similar to the foregoing, viz., the sub-
stitution of /i, ^, f^y &c. in the succession for x and the
determination of the coefficients one from another by means
of the equations thence arising gives ^o=F (fi) or, for bre-
vity's sake omitting the parentheses
• ^f-f^i^f-D {f~^)if-f^) (f-fx)(/^-/y
. Ff, ■ Ff,
^f'-f.) (/»-/,). • ..(/«-/,->)■
17
Several consequences follow fronv thts theorem ; 1st, If
F (x) be any rational integral function of x of the n*** degree,
andj^ any function of x whatever, it is easily seen that all
the values of the general expression for the coefficients after
^»inust vanish of themselves, giving -4.^^1=0, u^, + ,=0,
&c. to infinity, which is one of the most general and singular
properties of rational integral functions : (2dly), If F(x)=x*
zndj^(x)=:x% we get the series of coefficients investigated in
the last question: (3dly), If F(x) = any rational^ integral
function of x^ and/" (a:) =z x* we obtain a set of coefficients
proper for the resolution of such a function as
into the form
A.x-\-J,{x-l) x{x-\- 1) + &c.
which will often be found exceedingly convenient, and of
which we shall give.examples (See Ex. 9, 10, of this section).
(4thly), If F (07)= any rational integral function of a? and
y(x)= ±x, we obtain general formulae for the resolution of
any rational function into preceding, or succeeding values of x.
(5thly), If jp* (a*) = (]+«)', and/(^)=x, we get. the binomial
theorem.
These instances will suffice to shew that this mode of
developing F{x) is not a mere matter of idle speculation.
Other and still more extensive applications of it will shortly
appear.
. (6). To resold *', x*^ i*, x**, the even powers of x into
factorials where the preceding and succeeding factors occur
symmetrically.
This cannot be done, it is evident, by resolving them into
factors jr,.x± 1, x±:2, &c. because the degree of . any ex-
pression such as
^,T + -^,(.r~l)>(j + J)4-.&C.
* c
16
Such then being the values of A^, A^^ &c., we have
=^ A^x + A^ (x-l)x(jr + l) + &c.
V being equal to x*.
The law observed by the coefficients is not a little re-
markable. It extends too with a slight modification to cases
of much greater generality, and It will hardly be thought
irrelevant to the present subject to propose and resolve the
following problem.
(5). To develope F (x) in a series of factorial terms of
the following form
F{x) = ^ + A (*-/) + A, {X -/) (X -/,) + &c.
F{x) being any function whatever of Xy andyj, f^^ &c. par-^
ticular values of any other function f, corresponding to the
values 1,2, &c. of x.
A process exactly similar to the foregoing, viz., the sub-
stitution of fy, yi, f^y &c, in the succession for x and the
determination of the coefficients one from another by means
of the equations thence arising gives A^^F (fi) or, for bre-
vity's sake omitting the parentheses
ji "•yi jn ~yi
. Ff, F^
^ ""' (/.-/.)(/; -/.)-(/i-/»)V.-/i)(/.-/3)"(/.-/.)
(/.-/) (/»-/.)• • . .(/.-/._>)*
17
Several consequences follow fronv this theorem ; 1st, If
F (x) be any rational integral function of x of the n^ degree,
andj^ any function of x whatever, it is easily seen that all
the values of the general expression for the coefficients after
^.must vanish of themselves, giving iln^.,=0, -<4, + ,=0,
&c. to infinity, which is one of the most general and singular
properties of rational integral functions : (2dly), If F(x)=x*
andy(j?)=a:% we get the series oiF coefficients investigated in
the last question: (3dly), If F (x) = any rational ^ integral
function of x, and/" (a:) = x* we obtain a set of coefficients
proper for the resolution of such a function as
x.F{x*)
into the form
A^x + J,{x-l)x (x + 1) -H &c.
which will often be found exceedingly convenient, and of
which we shall give.examples (See Ex. 9, 10, of this section).
(4thly), If F (07)= any rational integral function of ;c and
/{x)= ±:x, wf obtain general formulae for the resolution of
any rational function into preceding or succeeding values of x.
(5thly), If i«'(x)=(] +a)', and/(x)=x, we get. the binomial
theorem.
These instances will suffice to shew that this mode of
developing F(x) is not a mere matter of idle speculation.
Other and still more extensive applications of it will shortly
appear.
. (6). To resold *', x*, i*, x**, the even powers of x. into
factorials where the preceding and succeeding factors occur
symmetrically.
This cannot be done, it is evident, by resolving them into
factors jr,^x±l, xd: 2, &c, because the degree of . any ex-
pression such as
^, X + -^» (x— 1)> (X + J)4-.&c.
* G
18
must necessarily be odd. The object may however be
• 1 3
accomplished by taking x ± -^ , ^±-9 &c. for the factors,
when we shall find
^ = -ii l + 10(2j:~l)(2:r-hl)
Id
+ (2j:--3)(2jr~l)(2a + l)(2^ + 3) \
i« = -L| 14.91 (2i— I)(2T+l) + 35(SJr-3)...(2^ + 3) +
-f (2j:— 5) (2j? + 5) \ .
For j:*"; — put (2j?)* = », and supposing -^(11?) = ^", and
y(v)=(2 V— 1)% we have - .
/. = l%/,«3%/3 = 5% &c.
Pfi ^ 1^ . ^/. = 3-, H = 5% &c.,
and we shall therefore have by tiie general theorem above
dertKmstrated (Art. 5. Sect. 2^)
ty=:^o+A (v-l*) + -rf,(v-l»)(v-3») + &c; where
l»~3' 3'— r'
^ _ . . . 1*^ . 3^. . . ..6^. ...
•- (l«-3«) (l«-5») (3*-~ r)(S'-5«) (5»— !•) (5* -3*) •
Now since (2 xf zzv . j:** = — • and we therefore have since
2**
and so on, the following final result, where ^ot Sec. have the
values above written
/
19
j;*»=^M.+^.(2ar-l)(2*+l)
In the same way may such a function as F (x*)^ any even
function of <v be treated.
(7). To complete the factorials
x(x+l) (x + 3), and (2 x— 1) (2x+l){2x + 5).
N. B. By completing them is meant reducing them to the
others in which the terms follow the ord^r of an arithmetical
progression.
Tfeey are respectively equal to
a: (jp + 1) + « (x -f 1) (« 4- s-), and
2(2af— l)(2a^ + l) + (2 j?-1>(2 jr + I) (? ^ + S).
(8). Reduce (2 a: + 37 to preceding values and (x+a)^
to succeeding ones of x.
(2 *- 1) (2 «-3) (2 x-5) -f 18 (2 «- 1) (2 * -3) +
+ 76 (2:r-. 1) + 64 = (2 X + sy
(x + ay = (a- 1)* + (2 a-S) (jr + 1) +(•»+ 1) (-r + 2).
(9)« Reduce j: (a* + 2)* to factorials in^which the pre-
ceding and succeeding factors occur symmetrically. The
application of the theorem in (Art. 5. Sect. 2.) gives
X (j:*+2)»=9 x + 9 (j?- 1) (x) (a:+ 1 ) ^
+ (*— 2)(j:-l)x(a:+l)(A' + 2).
(10). Reduce (x^+x) (2 a:*— 6)^ in the samiB manner
.(j' + ^) (fi J^ - 6)'= - 128 Jr + 56 Cr— 1) x (x-^l)
20
+ 424 (j;-.2)(:r-l)x(^4- l){x + i)
(ll). Resolve (x^-k- ly int6 the smallest possible numbef
of complete factorials.
(jr^+iy =i { 289+1140(2ap-l)(2i?+l)
-f 998(2 j7_S)(2x- J)C2 jr+ 1) (2 a: + 3)
+84(2ir-5),...(2^ + 5) + (24f— 7) (2jr + 7>{ ;
(12). Affect tt,+ i . Wx + 3 with u, and reduce it to
succeeding values of u,, N.B. By affecting it with u, is
meant introducing u, as one of the factors of the result. In
this and the following examples u, is understood to have its.
difference constant, so that
Then, m, + i.u, + ^= w,w,^i + 3Ai/,.
(IS). Affect u, «. 3 . M, + , with Uj,
(14). Reduce (w,_ ^^.(w,^.^)* to succeeding values of
tt*4-l ^x + 4 ^ 10A.M,+ , .... tt, + j
^ (15). Affect M,— i w, + 9 ^'x + a with w, and reduce it to
factorials consisting of u^ ^nd its ^succeeding values.
21
(16). Reduce u,u,^^ m,^, «*x + 4 ««* + 5> to complete
factorials.
(17)- Reduce u, i«^ + »-i x t*, + »+, «^* + m to
complete factorials.
a
4-(i»— «) (wi — «— 1) A* .w,. . . .w,4.^-_3
-f . . . .{iw— «). . . .3 .2 . 1 A*"""*M,. . . .M,4.„_,.
(18). Affect tt,— '2 w,_-i W;, 4. J with w^
(19). Reduce u* .n,^]^ to factorials in such a manner
that all the terms shall be positive.
+2 A* M,-_^ M, ^,4. 1 + 4 A* tt,w,4. J.
(20). Affect w, -4. 1 . . . . w, 4. » with w,.
«,4.j. . . .Wjp^-„ = ai,. . . .W^^,n_j+/lAM,. . .•ll,4.»^»4"
+ W («— 1) A*W,.'. . .M,+,«.3 + . . . w .n («— 1). . ..1 . A"*
(£1)« Retduce ii,. . ..ti,4.„ to factorials ending with any
value w^ + « + n,
Ug. , . . Wx 4* « = ^jr + i« • • • • ^* + « + »
in+ l).m
I
A«x + m+l-' ''^^ + «-*•»
, (w+l).»xw(iw + l) ,«
1.2
_ (n4- 1) n(n- 1) x /» (m -hi) (»i + 2) ^3 g^c. + Scc.
1 .2.3 ^ * * "^
The Investigation is Qasy if we enxploy the principle explained
in (Prob. 4. Sect. 2.) by assuming
+ -Bm^^^ + i-. ..Mj, + ^ + „ + &c.,
and making 1^^4.^4.1, v^x + m + a> &c* vanish in succewon>
which will give so many equations for determining A^ B,
Sec one from the other. The two last expressions are from
Emerson's Increments.
(22). AfFect u^j^^u^^^ with u, ,
(2S). Affect M^^^ ^^+^«',+ 7' ^th M„
+ (a/9 + a7+/97~a-./9-7 + 1) ^•m, + a/97 . A^.
(24). To determine the general law observed in these
reductions, or, to reduce the function
u^ u^ , 'U , X &c.
*+a ^ + /8 *+7
where the number of factors is n to a series of complete
factorial3 commencing with u, and proceeding according to
the succeeding values ^x + i* ti;r+9>&c*^« being as before
equal to a + Ax.
Assume jif^ + ji^u, + ji^u^u^^^-^ &c. for the series
and make .r in succession — - » "^ ( A "*■ ^y ' "" Cl "*" ^ r •
&c. so as to cause the factors u,, u^^i^ &c. to vanish in suc-
cession, and the resulting equation will give the coefficients
in succession just as in (Frob. 4. Sect. i\) Or we may pro-
ceed by supposing v = hx^ JF(v) mti^^^.u^^^f 8cc.
or, F(v)=i{v + u + ah){^ + a + /3h). ftc.
and /^ = — <j — (j: - 1) A,
and at once substituting these values in the general expres*
sions there given for the coefficients and reducing. In either
wa^ we shall obtain for a final result the following general
and useful
THEOREM.
Let S^ = 1,
S, =s a)3 + a 7+ /^7 + &C.
^« = « /5 7 X 8cc. ^
Then will
"x + a • «x+^ • ",+> . &C. = A- S,
+ !5lll' «,«,+ ! { A«o' . 5,_, - AV. 5._3 + &c, }
+ . ~\ ^' "x + i«x+, ( AV . S,,, - Sic. \ +8tc.
Ln — i
The terms a d" . S„ in tlie coefficient of — - — «„
A'o^. iS^ — A*(?* .5»«.i in that of — ^ u,u,^^ and so on
being of themselves equal to zero are for brevity omitted in
the respective series within the brackets, to which they belong.
Nevertheless to preserve the symmetry of the equation, they
ought^ if not set down> at least to be understood, being in-
«3
The investigation is easy if we enxploy the principle explained
in (Prob. 4. Sect. 2.) by assuming
Ug. * - • l^jr H- » ^ -^ **x -4- Hi * • • • ^jp + m 4- »
+ Bu,^^^^., , .M^ + ^4.„ + &C.,
and making ti« + m+i> ^^x + m + a> ^'C* vanish in succesfion^
which will give so many eq[uations for determining A, B,
&c- one from the other. The two last expressions are from
Emerson's Increments.
(22). AfFect «^ar + a ^x + /3 ^^* ^' »
(2S). Affect M^^^ ti,^^ w,+ y with m,,
M^ + a^ar+zB^'x+Y =«'*^* + l^' + «+(«+/' + 'y-5) A«;M, + l+
(24). To determine the general law observed in these
reductions, or, to reduce the function
where the number of factors is n to a series of complete
factorial3 commencing with u, and proceeding according to
the succeeding values u,j^^^ 1^,4.9, &c. t/« being as before
equal to a '•\-hx.
Assume A^-^ A^u,^ A^u^u^^^-^^ &c. for the series
and make .r in succession — - » "^ v J "*■ ^ J • "" \I "*" ^ # *
&c. so as to cause the factors u,^ u^+tf 8cc. to vanish in suc-
cession, and the resulting equation will give the coefficients
in succession just as in (Prob. 4. Sect. i\) Or we may pro->
ceed by supposing v zz hx^ F{v) mu^^^.u^^^^ &c.
or, F(v)=i(v -f « -r «*)(<^ + « + /?*)• *c.
and f,zz — <j — (j; -. i) ^,
and at once substituting these values in the general expres-
sions there given for the coefficients and reducing. In either
wa^ we shall obtain for a final result the following general
and useful
THEOREM.
Let S^ = 1,
S^ ss a-f)S + 7 + &C.
S, 5= a)3 + a 7+ /^7 + &C.
5'« = a /9 7 X &C. ,
Then will
7^x+a • '^x+^g • '^x+> • 8^.^- = ** 'S,
+ !CZlu.u,^.^ \ A^o' . 5.^, - A«^. 5«^3 + &c, }
The terms a o* . S^ in tlie coefficient of — - — «,,
aV. iS^ — A*(?'. 5»_i in that of — ^ w*w* + i and so on
being of themselves equal to zero are for brevity omitted in
the respective series within the brackets, to which they belong.
Neverdieless to preserve the symmeftry of the equation, they
ought^ if not set down, at least to be understood, being in-
34
eluded in the law of the other terms, and tJus remark is to be
considered as applying to all similar cases^ £!^ o^ being unity and
^ (?% A« 0^, A« o\ A^ 0^, A' <?*, A' o\ &c. respectively zero.
Some particular cases of the general theorem deserve to
be stated separately, as they afford' transformations which
which we may have occasion to use.
(25). To resolve m\ ^ ^ into succeeding factors affected
withv,. Here a=:/?=&c. = l : and
fgA.^^^^"^^A.--i-^^^-^^^^^^^AQ^^&c.|
ll 1.2 1.2.3 >
«x «x + 1 X
1 .3
y!L(!lzl)A'(^'~''^^'"^^^^"'^^AV^8cc.l
I 1.2 1 .2.3 y
'*"T7273^'''"*'^'''"^'^
p(,.,l)(n-2)^3,3^8,e.K&c.
C 1 .2 . 3 >
But this is not the simplest or most elegant form in
which this equation can be expressed. If we completes the
series within the brackets by inserting the deficient terms at
their commencement^ and then separate the symbols of
operation from those of quantity they will become
- A [ 1 - % + ^^^f^O*- &C.]- = - A (1-0)-
+ A» J 1 - 2 9 + "^"~^^ o- - &C. J = + A* (1 -0)-, 8CC.
lib
Now the reader will find it demonstrated in Att. 17.
Sect. 7. that the expressions A (1 — 0)% A*(l - 0)*, 8cc. Jure
respectively equivalent to
(- 1)". { Ae?**- A^o" + A'a« ±A»o*},
(- 1)». { A«<7* — a3o«+ q: A"©'* } ,
(— I)"* . } A' o» ± A"<?» } , &C.
or, inverting the order of writing their terms, to
— A«0*+ A*~'d". ... ± Ad"
4.A*<?»- A*~'<^ ± A^A &C.
SO that by substitution we shall have
» 1^. A*e>*-A"-'e?*+ ±^o* j^^,
M\4.^=ir+ ■ ■ • «* *tt,
^ A»(?*-A*--»(?»-f....qpA««>*^_. ^ ^^
+ &c.
The reader must not be startled by the employment
of as an algebraic symbol in such expressions as A(l — 0)"«
He will call to mind that this and similar expressions are
mere abbreviations and have no meaning beyond what is
expressed by their developement. The transformation in the
latter part of this problem cannot, however, be cbmpre-
hended without a previous knowledge of those more general
I)roperties of the functions A* ©* which will be hereafter
demonstrated, and is only inserted in this place t)iat things
relating to the same subject may be kept together.
(26). To resolve ti*, into u, and succeeding factors
18
nnist necessarily be odd. The object may however be
1 s
accomplished by taking x ±, -^ ^ .r±-, &c. for the factors,
Ziehen we shall find
x*^l\ 1 4.(2^^1) (2a:-M)},
4
^ = i{ 1 + 10(2j:--1)(2x-|.1)
lo
I
+ (2 j?--3)(ex-l)(2a + l)(2x + S) {
A« = J~{ 1+91 (2a:- l)(2ar+l) + 35 (2* -3).. .(2 ^ + 3) +
04
-h (2^—5) (2JP + 5) } .
For x*" ; — put (2 xf = », and supposing F{v) = t?'V and
y(v)=:(2 V— 1)% we have ~
/ = l%/, = 3%/3 = 5% &c.
^/i= l•^ J^/a = 3«% i^/3=5- &c.,
and we shall therefore have by ^e general theorem above
demonstrated (Art. 5. Sect. 2%)
v"=^o + -A» (v— l*) + -rf,(v~l')(v-3') + &c; where
. 1«* . 3»»
l»-3* 3'— P'
jl^ . ■ . 1'" . 3^. .. . ..^'V...
• <l»-S*)(l*-5') (3»— n(3»--5«) (5*— i»)(5»-3»)*
Now since (2 a?)* =: v • j:** = — , and we therefore have since
«-!•= { v^(t?)-l J 1V(^)+1 J =(2x-l)(2r+l),
and so on, the following final result^ where A^^ Sec. have the
values above written
/
19
X^-\ \ il.+^. (2ar-l)(2*+V)
ft
+ il,(2x-3)....(2i? + S)+&c. } •
In the same way may such a function as F [x'^)^ any even
ftinetien of j? be treated.
(7). To complete the factorials
!• (J? + 1) (* + 3), and (2 x— 1) (2 x- + 1) (2 x + 5).
N. B. By completing them is meant reducing them to the
othefs in which the terms follow the ord^ of an arithmetical
progression.
Tl^ey are respectively equal to
•r (« + 1) + « (x 4- 1) (x 4- 2>, and
2(2x-l)(2^+ 1) + (2x-.l>(2x+ l)(?^ + a).
(8). Reduce (2x + 3)* to preceding values and (x -I- a)**
to succeeding ones of x.
(2 X- 1) (2 x-3) (2 X -5) + 18 (2 «- 1) (2 x -3) +
+ 76 (2 x-1) 4- 64 = (2 X + 3)'
(x + ay = (^i- 1)* + (2 a-S) (x + 1) +(x+ 1) (x 4- 2).
(9)' Reduce x (x* + 2)* to factorials in^which the pre-
ceding and succeeding factors occur symmetrically. The
application of the theorem in (Art. 5. Sect. 2.) gives
X (x*+2)* =9 X + 9 (•»- 1) («) (x+ 1 )
#
+ (af— 2)(x-l)x(x+l)(x + 2).
(10). Reduce (x*+^) (2x*--6)' in the samje manner
.(x» + x) (« X* - 6)'= - 128 x + 56 (x-- J) x (x+ I)
20
-f 176(*-3)(j:-2)(3:- lXa?+l)(i?+2)(;«+3(+ +(«-4). .. .
• • • « I <i "f" *?yf
(ll). Resolve (a^+ l)^ into the smallest possible nambefr
of complete factorials.
(^+i)* =1 { 289+ 1140 (2 jf-1) (2 J? 4-1)
4- 998(2 j;—S)(2x-1)(2j:+1)(2x + 3)
+&4(2*-5) (2:r + 5)+(2 4F-7) (2x + 7>{ .
(12). Affect Wx + i • w, + 3 with m, and reduce it to
succeeding values of u,, N,B. By affecting it with u, is
meant introducing u, as one of the factors of the result. In
this and the following examples u^, is understood to have its'
difference constant, so that
u, = id + kx, A u^ = A,
Then, u,^^.u,^^:=: u^^u^^^ + Sku,.
(IS). Affect M,_3 . w^ + ^ with k,
(14). Reduce («,—0*-(w*4-i)* ^^ succeeding values of
ti
^x-hl • * • • "' + 4 "^ 10 A. M,^, .... W^-fj
, (15). Affect M,— 1 w, + 9 ",4.3 with Ug and reduce it to
factorials consisting of u^ and its Isucceeding values.
(16). Reduce w, w,+ , u,^^ «*^+4 w^ + 5> tP complete
factorials.
jinx. ti,.. ..Us^^-k-^hu,. . . .Wj, + 3 + 2A*w,.. ..M^^,.
(17> Reduce u, ii, + H-i x ii,4.«+, «*, + • td
complete factorials.
w,.. . .tt, + „_,+ (m— «) A.M, «, + „-.,
4-(m— «) (jw — »— 1) A* .w,. . . .i«,^^_3
+ . . . .{iw— «). ...3.^.1 A*"*"*w,. . . ,w,4.n~i.
(18). A£Fect m,— -, w,_ ^ m, ^. , with m^
(19). Reduce u^ .u,^^ to factorials in such a manner
that all the terms shall be positive,
+2 A* ?/,— 1 M, 11,4. ^ + 4 A* UgUgj^^.
(20). Affect M, + 1 . . . . w, + „ with m,.
t«,4.^. . . .M,^.„ = M,. . . .tt,^.n--l+«AWj,. . ..W,+ „-ia4■
+ n (»— 1) A*W,.-. . .M,^,_3 + . . . w .« («--l) 1 . A\
(£1 ). Reduce ti, . . . . u, 4. « to factorials ending with any
value tt, + „ + ^,
^* • • • • W* 4. » = ^4r + jn • • • • ^r + m + »
{n + l).m ,
— ' ' « ^* + m + 1 • • • • ^* + »• -4- »
+ ; — r ^ »* + m 4- a ••••** + » + »
1 .2
(» 4-1 ) n (yt - 1) X ffl (m 4- 1 ) (»i + 2) y &c. + &c.
1.2." '
The investigation is Qa$y if we employ the principle explained
in (Prob. 4. Sect. £.) by assuming
+ ^ «'* + «. + 1 Wx+m + » + 8lC.,
and making u^^^^.^, f/jr + m + a> ^'C* vanish in succes^iimy
which will give so many equations for determining A^ B,
&c* one from the other. The two last expressions are from
Emerson's Increments.
(22). Affect u^j^^u^j^^ widi u, ,
^x^a^x+0 = «^x w* + 1 + (« + y3- 1) All, + a /? . A^
<2S)- Affect tt^^^ ^^+^^,+7» ^th u„
(24). To determine the general law observed in these
reductions, or, to reduce the function
U U \U X &c.
where the number of factors is n to a series of complete
factorial) commencing widi u, and proceeding according to
the succeeding values u,^^^ u,^^,iic. u, being as before
equal to a -|- Ax.
Assume J^ + A^u^ + A^u,Ug^^-¥ Sec* for the series
and make a: in succession — - ♦""Cx'^'^y* ~\i"^^7*
&c. so as to cause the factors u^^ t^^+i* See. to vanish in suc-
cession, and the resulting equation will give the coefficients
in succession just as ixk (Prob. 4. Sect. 2.) Or yire may pro*
ceed bysupposing v = hxy Fd^) m u^^^ . m^^^, &c.
or, F(v)^(v -h u -r ah)(fi + a + fih). ice.
and /^ = — fl — (x - 1) A,
and at once substituting these values in the general expres*
sions there given for the Coefficients and reducing. In either
wa}^ we 3hall obtain for a final result the following geftc^I
and useful
THEOREM.
Let So = I,
S^ sa a^/3+7 + 8CC,
Sn=^ afiy X &C. ,
Then will
+ ^^u. { Ao. S,_,- Ao« . S,_,+Ao»,5,_,-&c. {
+ ^' «.«.+ ! { A«e«. 5._, - AV. 5._, + &c. }
The terms A ©• . S„ in the coefficient of ii„
^•o®. 5„ — A*o*. S,-.! in that of — ^ m,w,4.i and so on
being of themselves equal to zero are for brevity omitted in
the respective series within the brackets, to which they belong.
Neverdieless to preserve the symm(ftry of the equation, they
ought^ if not set down, at least to be understood, being in-
34
eluded in the law of the other terms, and tJus retnark is to be
considered as applying to all similar cases^ t^ o^ being unity and
1^0^, A«o°, A«^", A^o^ A^ o\ A^ 0% iic. respef^tively zero.
Some particular cases of the general theorem deserve to
be stated separately, as they afford* transformations which
which we may have occasion to use.
(25). To resolve u\ ^ ^ into succeeding factors affected
withv,. Here a=:)^=&c. = l : and
Alt'-. 1
ll 1.2 1.2.3 i
1 • 2>
5 "("-') A'O* - "(»-^)(«-g) AV-&C. ]
1 1.2 1 .2.3 3
1.2.3/ ^ ^
{^i?LLll^?^>^3,,^&e.K&c.
C 1 .2 . 3 >
But this is not the simplest or most elegant form in
which this equation can be expressed. If we completes the
series within the brackets by inserting the deficient terms at
their commencement, and then separate the symbols of
operation from those of quantity they will become
- A J 1 - - <» + _L__ie»_ &c. J- = - A (1-0)-
+ A» J l-2» + l^^Zlto'^ &c.] = + A«(l-0)", &c.
lib
Now the reader will find it demonstrated in Att. 17.
Sect. 7. that the expressions A (1 — 0)*, A*(l -0)*, 8cc. Jure
respectively equivalent to
(- 1)«. { Ai?**- A^o" + A'a« ± A»o* } ,
(- 1)». { A«o* — A3o»+ h: A»o'* } ,
(— ly . } A5<?» ± A"0» } , &C.
or, inverting the order of writing their terms, to
— A«0*+ A*~'d" ± Ad"
4.a*<?''-. a*~'<^ i A^A &c.
so that by substitution we shall have
^ A»o»-A*-'(?'*-f....qpA««>* _,^^ ,^
+ Y~2 ««*«*#4.i
+ 12 3 A"-'«,w,+i«^,-i..
+ &c.
The reader must not be startled by the employment
of as an algebraic symbol in such expressions as A(l — 0)*-
He will call to mind that this and similar expressions are
mere abbreviations and have no meaning beyond what is
expressed by their developement. The transformation in the
latter part of this problem cannot, however, be compre-
hended without a previous knowledge of those more general
properties of the functions A* c* which will be hereafter
demonstrated, and is only inserted in this place t)iat things
relating to the same subject may be kept together.
(26). To resolve u\ into u, and succeeding factors
* D
26
If in this we put n + l for n and divide by u, we get as
follows :
(27). Let Ug^a-^hx, u\=^ a + k x. To resolve («,)'
into t^', and succeeding factors^ ^^+ n &c.
'-*'^=(l)" ■(¥+")■
or, (ll,r = iT.i .(«<'.+ a)"
, flA'— hcL a a
where a z= =:- — -«
Therefore in the general expression (24) writing i*', for
u, and making /?, 7, &c. equal to a, we get as follows :
+ 8cc.
}
Now, it will be proved in Art. 17. Sect. 7. that the
series
37
C 1 1.2 >
I 1 1.2 3
vrhich the separation of symbols of operation from those of
quantity (as in the last example but one) produces from the
coefficients of the several terms Ugy See. and which have
for their abbreviations respectively
— A (« — of, + A* (o — oY, &c.
are equivalent to other series^ the respective abbreviations of
which are
A . . .„^« A»
(-i)»+» — ^^—^0-, (^i)»+* — :i—o%
(l+A) (1 + Af
(-!)» + '. ^^0%&C.
(1 + A)
the series themselves being
^ I 1 1.2
-fc «(«+l)-" (« + n-2) ^,^>
1.2 (»-!) i'
(- 1)« + *|a*<?*— ^A'^?'*
a(a+l)....(a+n--8)^.^.7 ;
1.2 (11-2) >
and the term a* itself is shewn by (Art. 17. Sect. ?•) to be
equivalent to ( - 1)* . -o"", or to the series
(1 + A)
(,l)4,>^^^A.-+....±"<"+^^"-("+^^^^^A".-l.
28
Thus vie arrive at the folloMring equation
'/» 9
which affords an indefinite number of different ways of trans-
forming any given expression of this kind. For example :
(28).
(1 + Af 1.2
SECTION III.
Exercises in the Reduction of Fractional Expressions
to Integrable Forms.
(1). Reduce -— — -— -— to an integrable form.
It becomes
X 3
(a? + 1) (« + 2) (x+ 1) (« + 2) (x + S) '
(2).
(JT + l)(ar + S)
J 1
(x+l) (a+2) (;v+ 1) (.:r + 2) (r-fS) *
39
(8). ^* •♦■ t ^ ^ _ S h
4 A^
+ — where w,= ^j+A*.
-%
(4).
(0? + !)*
(5).
J:'('«? + 2)(* + 3)(x+4) a?(x+J) •>?(^+ l)(J?4-2)
+ ^9 g7
a: (j^+3) a? (a? + 4)'
3jr-4 3 I
(4j^»~1)(2x-.7) 2* (2x-l)(2a:-|-l)
.IS 1 . 26
2 (2j:-3)(2a:—l)(2T+l) (2a:-5) (2x4-1)
+ 52
(2x-7) (2x+l)*
(6). Theorem (from Emerson's increments,)
1 nh
+ »(n-l)h* _8^.
This is immediately deducible from (Prob. £1. Sect. 2.)
by making iw= 1, writing «— 1 for «, and dividing the whole
equation so prepared by w^p. . . .w, + ».
(7). A rather more general theorem (given by the same
author) is the following :
1 1 («-wi)A
which may be proved by a process almost precisely the- same.
30
These two transformations are not without their use in
facilitating reductions of a certain class. For instance,
(8). To teduce to an integrable form,
1 1 2 A . 2 A'
therefore
1 1 2A . 2A'
W, — lW, + a «,-.i«^x , W,— 1 «^*+l «** — 1 ^x-k-
(9). To reduce • in like manner
1 1 3 A . 6 A-
r
6A»
and
3 A . 6A«
+
6A»
31
SECTION IV.
Exercises in the Integration of Equations of
Differences.
( 1 ). To integrate the equation
The complete integral is
Ug= (C 4- a?) xl .2 X
C being an arbitary constant quantity.
«»
(2). u, + ^—pa^*u,:=:qa
^ \ -- ap
(5). («+irUx + .-«ti, } =a'
{x+ If
In this Example the integration of ^ cannot be per-
formed in finite terms, unless we express it in a series the
number of whose terms is variable. This we have done, and
in many cases (as we shall see) such a result is useful and
satisfactory*
(4). w, + , ^pu,^^ +9W>=0
32
'C and *C being two arbitrary constants*
(5). !/, + £- ^1 (x + 2)w,+ .+*(x+l)(x + 2) f/,=
«,= 1 .2 :r pC.a' + *C./3'}
a and /? being the two roots of V + "a « + * =: 0, The inte-
gration is performed by assuming
(6). «,+,+ a p"Mx + ,+*i»** «,=«?'./>''
a and fi being the roots of
^ a b
("7). «, + «+ ^•^(^+lK+, + ^.^(*) .^(j:^+ i)«r=^,
(a:) being any function of x whatever ;
«,=5P0(*- l)x
t «' l^'.FiPix + 1)5'
P0(J?) denoting ^(1).0(2) ^(x), and a and /3 being
roots of tt"* + a 14 + * = 0.
(8). «, + 3+ «y «** + .+ 6 P''«, + l+ i^jB^'WrSi^O
«> ^> y> being the three roots of
t4» + --«*+ — tt + -s = 0.
38
(9). «,+ «+flfpX + i.-i + */?'' «* + »-«+ kfu^^siO^
a, /3f. . . ,v being the « roots of
(10). H, + ,+ aM, + ,+ *tt,=X^,
a and /3 being the roots of «* + fli/+*=:0.
(11). A««,=X,, ,
(12). Au,+ A««, =:42w,
i^, = >C.7' + *C.(- 6)*.
(13). (r + 3)» . «,+ , - 2li±4^ «,+ .
is to be integrated by the help of a given particular integral
x+ 1 \x + l/-
(14.) u.+ ,-2m%+ l.= Oi .
_ \
The substitution tb be used is u,ss cos v,.
* E
34
(15). «»,+ ,- 4 «*,(u%+ 1) = 0,
The substitution is », = >/(— 1) • sin v^
(16). «',-4«.«,+ , + a(2«, + «,+ .) = 0,
/
H Q V *^ 1
Substitute -< . — ' for u, and again^ in the resulting
equation^ — - cos k;, for v^^ when it will assume an integrable
form*
(17). «%+ ««,«.+ ,- a(4»,-u,+ ,)=:0.
Substitute 2 a . ^''^ ^ for «,
2 V, — 1
(18). i/, + ,w,+ i«* = «(w* + « + «, + i+ «*)i
Assume u, = v^(a) . tan v, then will the equation for deter-
mining V, be
whence finally
!/,= v^ (.1) . tan } -C . cos^^ + *C . sinl^} .
C 3 8 3
(19). The same substitution will be found to succeed in
the equations
w* + « «,-». . tt, rr fl (w, + a — tt. + . - w,) s 0,
w* + ««, + i«** + «(«*-»-• + «^«-».i- w,) = 0.
(£0). To integrate the equation
w, + I », - a («,+ 1 - w,) + 1 = 0. .
Laplace, Jwrw de PEcole Polytichniquei Cah. 15.
35
Differentiate it relative to x, and it becomes
(« + W.+ i) — — (« - u,) — ^ = 0.
» ax ax
But the proposed equation gives
a , >
«* + ! - w,
80 that by eliminating a^ we find
and integrating
1 +»%+. ^ 1+1*%
^ being a certain function of a to be hiereafter determined.
In fact) since both this and the proposed, are each of them
complete integrals of one and the snme differential equation,
the one can be nothing more than a transforb:)ation of the
other. Now this latter is equivalent to
dUg _^
du,^
because f t±L. is the same function of w,. , that
^ l + w%+.
^ is of x\ This equation is immediately integrable
(relative to the characteristic A) and gives
rJjiL^^Ax + c
Q being an arbitrary constant. To determine the function
A of a we first have to assign f i— . Now this is arc
^ */ 1 + w%
(tan = I/,) or, tan "" * (i^J, hence
36,
tan—' «, = -4a? + C
u, = tan (A x + C)
therefore, w, + , = tan (-4 x + C + -4)
^ ti, + tan
"" 1— w, .tan A '
But the proposed equation gives
a
W,+ , = J^
1 — M, . -
a
which compared with the foregoing gives tan jtf s -> or
A = tan""V- .^ The complete integral then of the proposed
equation is
u, = tan<a:.tan-Yiy+ c|
which, although in appearance transcendental, is easily freed
from that form, and reduced to a particular case of the inte-
gral found in the Appendix. Art. 386. of the more general
equation there discussed. The same method applies to certain
other equations, for which see the author cited.
(91). To integrate the more general equation
The proposed equation gives
but by /Art. 28. Sect. 1.) we have
A tan-* u, = tan-^ ( "^ — ")
37-
Therefore this becomes by substitution
A tan*~*w, = tan""'
^M
and integrating
tan-' «,= C + Stan-* i
or taking the tangent of each member
w, = tan< C + Stan-* i| ,
The preceding is a particular case of this, a, being there an
absolute constant.
SECTION V.
Exercises in the Integration of Equaiiqns of
Mixed Differences.
It will be necessary in the following cases to adhere to
the notation of partial differences we have before employed
(See Appendix. Art. 357, 364.) viz.
L.U -ia^x All ^^^'.y
^ ^^ ''>'■" dx ' rfy "^~"^'
and so on for the higher orders of the differential coefficients,
thus
\dx) \dy/^'''^ d^.d^'
The same mode of referring the symbols of operation to
their proper independent variables may also be conveniently
extended to the characteristic A, as follows :
38
A A
(A \-/^ A V
(^y "'+-"- tC^x "'+—■"+'"•
These two different modes of varjdng the independent
quantities x, y, may occur together^ as in the expression
Aar c^y ' a.y ay
and others of the like sort, and it is manifestly a matter of
indifference in what order the operations denoted by A and
d are performed. Equations of mixed differences determine
the form of a function by assigning a relation between these
derivatives. In Appendix. Art. 387. we have considered
mixed differential equations with one independent variable.
We shall here give a few instances where more than one are
involved.
(1). To integrate the equation
d
dy
«*+!,, — 3- «)r,y
This gives u,.^ zz (^} * (2/)*
ff> (y) being an arbitrary function of y. The reason is evident ;
for if we take «o,, = (y\ we have 1/, ^ = — ^ (j/), whence
ay
we derive t/,,,= (-r-J ^ ^^* ^"^ ^^ ^^'
(2). J^uppose — «*.y= « • 7- ^s.3,y
LikX ay
39
Assume u^^^^a'.e •.v,,y»
v^
and
alsoy
-if . .
^' + i,y = ^7r:^'.y
The equation then becomes by substit^ution and reduction
dy
which has already been integrated, and thus we get
^ denoting an arbitrary function.
. Given «^* + i,» = ^ • T^^ ^^y*
(3)
Assume Ug^^a' • t/^.y and we find for the integral
Assume, u,^y^f *Vg^yy and we get by substitution and
division of the whole by p'y
40
In this equation the sum of the indices x+2i ;r + i, .r,
below the v in each temiy and the corresponding exponents
0, 1, 2, of the symbols of diflFerentiation -— is the same : if
. dy
then we suppose
we shall have
and the whole equation is divisible by this function, leaving
p* — ap 4- ^ = 0,
to determine p. Let a and /3 be its roots^ then since the
proposed equation is of the first degree^ and either
«' . (^ / *' ^y^ ^^ ^' * \^/ ** ^^ separately satisfy it,
their sum is the general expression for u,^^,
0, (y) and 0, (y) being two arbftrary functions of y.
(5). ^' + »..y~«3-;«^' + »-..y + *(^) u,+n^^,,- ....
Let a, /3, 7, &c. be the roots of
then
*i (j/)> 0« (y)j 0« (j/) denoting n arbitrary functions of y.
•l'
41
But here a remark of considerable importance offers
Itself. It will immediately be observed that the process by
which the above two equatiotis are integrated is entirely
independent of the nature of the operation denoted by — ,
ay
It rfiight have been any other, and thus we might by the
same process integrate
or yet more generally
where v^*.^ denotes any linear combination of the differences
or differential coefficients of i«,,, relative to y, of whatever
order we please, nor does its generality stop here.
(6). To integrate the equation
VAa^y '^ ^x dy ^dy^ "
Assume M,^yS=a' . ^ « .v^.y*
a
and we get by substitution, and division by a' . ^ 1^,
+ -J { o*— aa + 6 I v,_,.
42
Let us for a moment suppose v,^ ^ = f -~ I (y), then
it is evident by the preceding problems that this equatioir
will become
= («*— «a+*)v,-|.a,^— -^.(ft*— ^a + ft)»r+i,.
+ —(«'-« a +^)t?,,y:
a»
the factor o* — aa + ^ being found in each term, if this be
made to vanish the whole is satisfied, so that provided a be
assumed a root of the equation
any error we may have made in our value of v,^^ is corrected,
and calling a and /S its two jroots we see that
each satisfy the proposed equation, so that
. . . . ± fe . I --- I /^^ ^ = 0.
A process similarly conducted will be found to lead to the
following result :
43
Where a, 0, 7, file, are the n roots of
and ^, {ji)y . . . . 0» (^) are as many arbitrary functions of y,
y
(2/) and >^ (2) denoting arbitrary functions of y and z.
(8). .^,,,, = «(i^) t._,„. + ^(^^) u_,„.
SECTION VI.
Exercises in the Summation of Series by the Inte-
gration of their general Terms,
As the integration of any function leads directly to the
sum of the series of v\hich it is the general term (Appendix.
Art 389*) the following examples may be looked upon as
exercises in that part of the inverse calculus of differences
which relates to the integration of explicit functions. To
sum then the following series,
(1). I. 2. 4 + 2. 3. 5 +3. 4. 6 + &c.;
S, denoting the sum to x terms^ we shall find
.. a:(a:+l) (x + 2)(S a: + 13)
o =^ — - ' ...... - ■■. ■■ ,
12
(2). 1* + 3* + 5* -f . . . . (^2 x-iy == *s;
44
(3). 5,= l+2/?4-3/?* + 4/>'4-&c.
S - ^ - P' ( ' + ^ — ^ p)
Sum to infinity (when p< \) =: S =
(1 - py
10 14 18 . o
(4), . + + + &c.
^ 1.2.3.4 2.^.4.5 3.4.5.()
s, = ?- ? . s = ?.
3 (j:+1)(j? + 3) 3
(5). ,.w. ■ >.^ +
V2 (1 + -s/S) (1 + ^/2) (2 + v/2)
+ I — + &c.
(2 + v/2) (3 + s/V
S. = — £ _ 5 S = J- .
(6). 1 . 3* + 3 , 5» + 5 . 7» + &c.
£, x(6x*+i6 x* + 9 x—4)
o- ^ ■ •
3
(7). 2. o. 8 4-4. 8. 14 + 8.14.26 + 16.26.50+ &c.
o 36 .23'+ 84 . 2«'+ 56. 2'- l7fi
'^' = ^
See Appendix*. Art. 374.
(8). _i_-_9_ + -^i_-_lL_+&c.
' 6.7 7 . 29 29 . 7y 79 . 245
S -J i . s = —
' 20 4 J2 + (- 3/ j ' 20 ■
45
,f^^ ^ . 3.19 , 5.19* .«
(9). ' 4- ' + Z + &C.
a.5.8.il 5.8.11.14 8.11.14.17
S, = ^-^ . - — — See Ex. 11.
li7(3j: + 2)^3JC + 5)(3j: + 8) <^lOO
/.,^^ 2 5.11 , 8.11« ^
(10). + — &c.
1.3.5 S.o .7 5.7.9
S, = -5 ; — tll}y . See Ex. 11.
24 8(2x + l}(2a: + 3)
(11). To sum the series
U^U^U^. . . .W»,— I W» W,.. ..U^ lift Wj W«+i
whenever it can be done, and to determine the condition
which must be satisfied to render it practicable^ (i£« ^u^^u^,
&c« forming an arithmetical progression).
The(r+ I)*** term is
U,U,j^,..
... .t^jr + M — \
Now the difference of a
function
A .s'
Ug, . ,
is easily found to be
• • • •^x ■+" W """ 5
A{su,
— w,4.«_,)x'
Ug. . ,
. • • ti, J. I„ ^ 1
For ti« substitute ^i+A<r and the numerator becomes
(J. \{s-\^a -(iw-l)A} -il(x-i)Ax)/',
which compared with {p -V qx) /*, that of the function in
question gives
4(»
and thus we get for the sum of the series
and for the equation of condition
P ^^
m — 1
q h s — \
also when / < 1
« =
(1— /)^.Wo ««.-,
The two series immediately preceding this example are
particular cases of this.
(12). +-- -f + &c.
^ 2.3 3.4 4.5
2 4 i: — 1 .
S, = - + Z . . 4'.
:3 3 r 4- 2
/lo^ 1*.9 , 2».9" 3^9' .
<^'>- s— +T?^■^■TT9■•'^'•
32 C 2 a: 4- 3 >
(14). To determine in what cases the function
ip + qx -{■ rx*) s*
is integrable, and in those cases to perform the integration.
Assume for the integral
W* ^» 4- m —
and by comparing the difference of this with the proposed,
we shall find
47
{s-\)h \h s-l y (s-i)h
and for the equation of condition
p-9.(l) + .(fy=
Whenever this holds good, the proposed function is
directly integrable the integral being as above (a).
If we resolve the equation of condition (h) with respect
to J, two values of s (real or imaginary) will be found which
render the proposed function integrable. This may be yet
farther extended, and by a process of the same nature the
foUovnng theorem may be proved.
(15). Theorem. // is always practicable to assign such
values of s, real or imaginary ^ being the roots of an equation of
^ the n% degree that the function
(a + fi X 4- 7 X* + 1/ X°) . S'^
Ux Ux -h 1 ^x -H m — I
shall Be' directly integrable^ a, /?, 7, . . . . i/, being any given quan-^
titles y and Ux being of the form a +h x.
(16). To integrate the function
9J
(hx -^ a)(hx -\- a -{- h)^
or to sum the series
i! 0L±1\\
(a + 2 A) V a y
(a +A)
2^ . (l±i)* +&C.
{a^lh)(a -f ,S K)
48
This function satifies the equation of condition (b) of (14),
and we shall therefore find for the value of the integral, or
the sum of the series to x — l terms
^ a^ h{x- 1) -- a /^ a + h y
A'(2£/ + A)* hx -^a \ a / '
or, determining the constant, and writing x+l for tf,
' l?{^aJfh)\ A(A? + l)-j-aV a / i'
This comprehends as a particular case the two series
summed in (12) and (IS). As series of this sort are not
without their interest, especially when considered in an ex-
tended point of view, a few more cases are subjoined by way
of farther exercise.
1 .3 3 3
(''^' ---^--+ltl"5^ + 7f7^S^« + ^^-
4 1 (2 x+l). 3^5' 4
Q 1 4 1
(18). --^ X i + -— X -- + &c.
S,«l~ ! , S^\.
[X + i;.2'
(19). — — X i 4- ^ X - + ^ X -L + &c.
^ ""^ 1.2.3 2 2.3.4 2-^3.4.5^2'
5^1 1 e^l
' 2 (JT + 1) (X + 2) . 2' * 2 '
(20). l+2+S+....a: = l^l±i).
2
(21). i« + ?« + ?H ra- ^(^+ 0(^ + 2) *(ftl)
3 2 ■
49
(22). 1^ + 2> 4-S» + .... x^
^x(x + l)(x+2)(x + S) x{x+l)(X'\-2) , x(x+l\
4 . 1 2
_ / j? (X + i) y
2
(23). l'*-f2'' + 3«+....^ = iS,
t - l.«2 1.^.3
• • • • tc A •
4?(d?+i),..,(j?+>i)
1 . 2.i ..(« + !)
$ee Equation (a) of Art. 26. Sect. 2. where A= 1, li^sj^H- 1.
The same siuns may abo be exhibited a^ follows ;
^(^ + 1)
(24). 1+2+.. **
i.e
(25)
. i« + 2'+....^'=i{^-^-i-^ +
(2 x-l) (!2 x+\) (2x + S) i
3 ^
^6). P+23+....^=<izi)i(i±i^£±!> +
4?
^a?t^ + l)
1 (QiX + 1
(27V l* + 2'+....a:« = ^Jf_jI^ +
^ ^g (2a:- l)(8y 4-l)(gJ +3) ^
3
, (2j-3)(2x— l)(aa? + I)(gx4 -S)(2j; +5)
+ -.
#
o
50
(28). 1S.U .5- (^-2)(^-n-...(x + 3r ^
6
. - (j: - 1) (d? + 2) . J(j? + 1)
'f- O , • . <^ , . -4- ■
4 • 2 •
(29). Ingenertl} l"*"^* + 2" + ' + ..^. «*"+' =
_ . x(x + l) , (j-l)....(x + 2)
-^o g— +A 4 ^ +
6
where the coefficients are those given in (Art. 4. Sect. 2,)* an*
1
!«» +2*»H- ^** =
r^ g^-)-^ r J (2^- 0(23:+ 1)(2j: + 3)
^ - ^, + IV 4, - !• .3^ il, + &c.
the coefficients here being those determined in (Art. (k
Sect. 2.)
(30). 1 +a-|-5 + . . . . (2 or — 1) == a*.
(31). V+3' + 5* + {2x-\y=z
(2 X— l).2a:.(2J? + l)
= 6^ •
(32). P4-3'H-5*+ (2^-^l)»= 2x*-x\
(33). The general expression including all these is
r-|-3'*+5*+ (2:r- !)» =
^ '^ C 1 1 1.2
(2 a? H- l)(2:r + 3) - 1 . 3 . «
. + occ.
2
51
(34). (a^hT^(a+2kT+ (a+3 hf+ .. . .(a + xhf=-
= «.*+w%-f M%
S.
"■^ ^ 1 1 ' * 2 1.2
3
Expressions for the sums of the powers of the natural
numbers were first given by Wallis in his Arithmetica infini-
$orufn for the. purpose of applying Cay^l^erius's method of
indivisibles to the quadrature of curvilinear spaces whose
ordinates are rational integral functions of their abscissae.
Their theory was treated in a ir^ore genejal way by John
Bemouillij and after him by Euler, to whom we are indebted
for the general theorem for the expression of 2 </, in a series^
whose numerical coefficients (from their identity, with those
found by John Bemouilli in the case of 2 x"^) he called by the
;iame of that Geometer* The expression^ given in the above
and sortie following examples for these sums are different
we believe from any yet noticed^ and seem to be the simplest
their nature admits*
(35). cos ^ + cos 3 a+cos 5 ^ + jcos (2 x— 1)0 =
sin X
sin
. cos j: 0. (Append. Art. 373.)
(36). sin a + sin 3 4-sin 5 0+ sin (2 ^- 1) a ==
sin j: .
= — : — T- • sm X 0.
smO
(37). cos + cos (0 +h) 4- cos (0 +2 A) -I- ... .
sin f - A y
4:08 1 a +(^-1). A} =^cos<e +^!Lllh\ . — yf >^ .
52
(S8). sin e+ sin (0 +A) +sin (tf +2 A) + . . . ,
sin (~ kj
' -GO
(39). 1 . (cos ey + 2 (cos 2ey + 3 (cos 3 ^y +
,...^(cos^a)^ = iii±i^4-
4.sm d - 4v 8in^ y
(40), ^ ^ .cos >► 4- sm S ^ . cos 3 >^ + . . . * . .
+ sin
2 sm
(41).
I
+
cos 6 . CQ$ 2 ^ cos 2 ^ . cos 3 d
1
III n
cos 3 ^ • COS 4 ^
^n + ^C,
^ tan (j? + 1) ^ — tan ^ c ^a^a c . . v
S=z, — ^ -,-f- 5 See (Art. 4. Sect 1.)
sm V
I
^ sin ^ . COS 2 ^ COS 2 ^ . sin 3 6
-^ +
+ ^
I
sin 3 ^ . COS 4
- &c.
53
' cosfl.sin 2(ar + l)9*^ ^ ' s&fl
: ... . •• >
(4f3% + — , — ^ H-
^ +&C.
sin 3 • sin 4 ^
« — -^ cotan (j + 1) 6 + cotang ^
(44). ' ■ • W^ . . — ^. ^ .. .. '^ .
• »
..\
cos . sin 2d sin 2 . cos 3
C0«5 5 6.$!n4 « '
g _ (- ly-^' H-cos2^i?4-l)0 tan 6
(45). 5 ^ i __1 : ■ , •
sin ^ . sin 3 e sin 2 ^ • sin 4 a
sm 3 9 • sm 5 ^
2 . cos ^ C sin ^ . sin 2 "sin (j? + 1) ^ . sin (a? + 2) ^ 1 '
See (Art. 20. Sect. 1.)
(46). ■ ■ ' ' ' ■ ■-■ ■ ■' +
cos ^ . cos 3 ^ cos S ^ . cos 4 6
I t «i w •
cos 3 • cos 5 6
- &e:
' 2 . cos 9 icos e .co^2 cos {x+ 1)6. cos(a; + 2)6 i'
See Art. 22. Sect. 1 .
54
(47). tan e + |tan - + J tan 1: + &c.
See Art. 14. Sect. !•
5, = -i^ cot -^ - 2 . cot 2 ^,
' 2»— » 2'""'
Sss I ^%. cot 2,0. See Lacroix, Translatiooi
y
Art. 57.
(48). (tan ey + i (tan |y + (j tan j)* + 8tc.
•-SV 2-; -^ (tan ? ^r /2'-.tan-£-y'
o ^ 8 4 _ 1
3 (tan 2 6)* fl* *
See Art. 15. Sect. 1. Also L^croix. Trandation, Art. 57.
(49). tan e . (sec oy + (^J tan -) (i sec -) +
+ (itan^)(i8ecjy + &c,
cos
S 2'-' _g^ cosSfl
(2— .dn^y (sinafl)'
*=i-*'(Sl^' SeeArt.l6.Sect.l.an4
Translation, Art. 57.
5fr
f, - 2' . f!l±i'_ l±i i (Art. i. Sett, I.>
(51), -^^—+—- +-y-- +&C*
^ ' «• - 1 a* - 1 a* — I
S, = i ll±i - "4^1 J (Art. 6. Sect. 1.)
(fi2).
a
^ + ^4r-- + ^-^ + -r-^. + &C.
sin sin 2 . sin 4 9 sin 8 9
5, = cot| ^ cot 2*-' ^; (Art. 18. Sect. 1.)
(53).
(
2 . cos - )
2/
S.=
(6\* • ^\
4 . cos ^j IS . cos - )
-^ +&cf.
1
1
(sin By /-__ . e\^
(2' . sin ^,)
^^{dw^'^'' (Art. 17. Sect. 1.)
(54), sin ^ (sin S) + 2 sin - (sin -.)[ +
+ 48m-{sm-) H-&c^
4 V S^'
iS •= i ^2' . sin sin 2^^
S = 5^!llLii. (See Art. 8. Sect. 8.)
2 4
^ 56
(55). (sin ey + 4 ^sin ^A* + 4^ . (sin -^ + &<i.
S. = (2'->sm^ -(-5-)v
S « fl» - (!^LEi)*; (Art. 9- Sect. i.
* * * * ' • .
tan r) + 2 tan ^^tan -^ +
+ 4 tan -(tan-) + &t.
'■ ' ■ ■■ . ' ' ' ' ' ' '
S, = tan — 2' . tan -J ,
S = tan - P . .(Art, 11. Sect. 1.)
/ 1 \* / 2 \' / 4 \
I .
8
+ /_^> + &c.
\co3 8 e/
* \ sin 2*6/ sinr
(58). To assign the value of the continued product^
P, = tan e (tan 2 ^)* (tan 4ef... .(tan 2' ^r.
If we sum the seties ^
log tan G 4- i log tan 2 + j( log tan 4 ^ + &c.
to jr+1 terms by the help of (Art. 19, Sect. 1.) and theri
transform the logarithmic equation into an equation of factors
by the well known property
d . log ^ + 6 . log B + &c.=log {A' . JB* . &c.)
57
we shall find for the value of P,
p _ 4 . sin ^
(2 sin 2' + * ey
and if P be the product to infinity, P = 4 . sin ^.
(59). To sum the series
tan-' !: + tan-* ^
1 + 1 + 1* 1 + 2 + 2*
+ tan-^ 1 +8cc.
1+3 + 3*
or, as it would stand in the ordinary notation,
arc Aan = — . "^ + arc (^tan = ^ + &c,
V 1+1 +1V V 1+2+2V
to X terms and to Infinity
5, = r — tan-* ^ JL-^ ; S = '^i (Art. 25. Sect. 1.)
Vj?+1/ 4
' 4
(60). tan-* -— - + tan-* — ^ + tan-* --i— + &c.
^ ^ 2.1* 2.2* 2.3*
S, = ^-tan-^— i ; S = ^ ;
4 2 J? + 1 4
Deducible from 26. Sect. 1. by taking h+xess2x+l.
(61). tan-*i + 2 -tan"* i + S^tan-*^-^^ + &c.
the progression of the denominators being
4 + 3, 4.8 + 3.2, 4.8* + 3.2', &c.
5.=2' .tan-' ~-~; S= 1- 7 ; (See 29- Sect. 1.)
2* 4 4
58
(62). tan-* i + tan-* 4; +tan-*JL +
4 13 27
+tan-*_ ^
3 x-i-5 X*
5, = tan-i i^ tan-i — , S=tan-*1.
2 5x + 4 2
(63). To determine in what cases the function
tan-* ;
p '\- qx + rx^
is immediately integrable^ and in such cases to sum the series
tan — * + tan-* -— r* +
P -^ q •\-r jt7 + ^,2 + r.2
+ tan— 1 + &c.
/> + §r . 3 + r . 3'
Let the function in question be compared with the second
member of the equation in (27. Sect 1*) and we find
besides which there remains an equation of condition to
be satisfied, viz.
^* — r* = 4? (p r — ) ) ; (a)
Whenever this equation then holds good, we have
Stan-* ^- = C- tan-* \ ? \.
p + qx + rx^ tq-^r + QrxJ
The series 59, 60, 62. are all particular cases of this,
and their general terms will be found to satisfy the equation
(a). Other examples are the following.
59
(d4f). Let the general or x^ term of a series be
1
Ug = tan ■" '
lOx* - 24x + 12
S, ==tan->(7) -t-tan^>(lOj:-- 7); S = - +tan'-»(7),
2
(65). Let u, = tan-- » — -^^ ,
34 or* — 8 J? — 8
then
S, = tan-' (34 J? + 13) - Un-»(13); S =tan-» — .
(66). Let«, = tan-^-^^^--j^-_^,
then
S, = tan-'(74a: +31)- tan-*-(31); S =tan-* — .
31
(67). Leti., = tan->^^^^,_^^— 5,
S, = tan-*(26 a:+5) - tan-' (5) 5 S = tan-' i .
The series (47, 48, 49, 50, 53.) are due to Mr. Wallace,
who gave them under a somewhat different form, among a
variety of similar ones in a paper communicated to the Royal
Society of Edinburgh in 1808, as formulas of approximation
to the arc of a circle (when continued to infinity) to which
purpose their rapid convergence, even in the most unfavour-
able cases, well adapts them. In fact we have by (47)
1 = ("2 . cot 2 ^ + tan 0) ^ \Uxi\Q + &c.
60
in which if we notice that 2 cot 26 + tan = we have
tan
the reciprocal of the arc, expressed in the form delivered in
the paper alluded to. , The series (50) when continued back-
wards by writing — a: for x give3
X
I
■^ X
1
a«
— 1
, 1
a^
_
1
i
a*
— 1
^« •
H
+ ..
1 • ^ • •
2
J
+ 1
4
a^
+
1
— jr
2'
— .jr
+ 1
_ ^
+
1
a"
+
1
a
—
1
a* /r
— — *
1
- 1
and this sum, when the series is continued to infinity will be
found to reduce itself to
a -^ 1 2
•— —— »
rt — 1 log a
This expression is accordingly given by Mr. Wallace in
the same paper, as affording means of computing the loga-
rithm' of an insulated number (a high prime for instance), or
at least its reciprocal, at once. It is true the operations are
laborious on account of the multiplied extractions of roots
and decimal divisions they require, but diey are not on that
account less valuable. Regarded in the light of elegant for-
mulae in the inverse method of diffejrences, these series assume
a higher rank in the scale of analytical estimation, in propor-
tion to the difficulty of that field of research, and the little
reason we have to hope for any fai^her progress in it. For
this reason, I have added the series (52, 54, 55, 56 , 57.)
which are of a similar nature, but have not been noticed by
him. Of these (54, 55, and 56.) afford in like manner, for-
mulae of approximation to the arc of a circle, viz.
^ =i sin2a +
2
+ C> I sin 9 . (sin ^)' + ^i sin t (sin J)' + &c. I
e^
61
sin 2 ^\'
=(Pfi)^
+ I (sin ey + 4 (sin |y + 4* (^sin ^y -I- &c. ] ; (a).
= tan 0— < tan ^ ^tan -^ + 2 tan « Aan - ^ + 8cc. > -
These all converge with the same degree of rapidity after
a few of the first terms, viz. nearly according to the powers
of ^, but for actual computation, the formula (^i) far sur-
passes the rest in convenience. They differ from Mr.
Wallace's in giving the immediate values of the arc and its
square instead of their reciprocals.
The continued product (58) is due to Mr. Babbage : the
summation of the series of reciprocal sines (52) may be
obtained from it, by taking the logarithmic differential rela-
tive to 6 and vice vers&y the latter may be derived from the
former by integration : The method^ however^ in which we
have here presented them has the advantange of exhibiting
the principle on which all transformations of the same kind
ultimately depend.
The series (59) and (60) are noticed by Euler in the
Comm. Acad. Petropol. ix. 1737. p. 234., as well as by
Spence in his Logarithmic Transcendents. By the former
they are given as particular instances of a general formula of
great neatness, at which he arrives by a kind of tentative
method, but which may be obtained very shortly thus :
Since A tan-' i£, = tan-^ ( -^ ^ ); 28. Sect. 1.
therefore we have
2 tan - » ^^-'.- = C -h tan - ' «,.
I + if^ Nr + 1
62
Thus we get for the sum of the series
C. 1 + u^u^y C J + WjW,>
the following expression
tan ~ ' «, — tan "~ * ?/o .
Which is in fact Euler's formula^ for u^ being a function
of X of a form perfectly arbitrary, its particular values w^j Wj,
t^si. . . .«, to any extent we please, may be looked upon as so
many arbitrary and independent constants, and may be repre-
sented by separate letters, o, ft, r, &c., which done, a very
trifling reduction will give the formula in question. These
series of inverse tangents in which the numerators of the
fractions under the characteristic tan — * are unity, and the
denominators integer numbers (as 59, 60, 61, 62, 64, 65, 66y
67,) are extremely remarkable on account of the facilities
they aflFord for extending the integer evaluation of the func-
tion tan"~*(j:) or as Spence denotes it *C(j7) . (64. . . . 7)
have not, I believe, been noticed, nor has (6 1 ) which is not
included in Euler's general formula, but may be derived
without difliculty by a method similar to that by which that
formula was originally obtained, from the following equation
tan — '- + tan — ' — =s 2 tan-* — ; (a).
which the reader will have no di£5iculty in verifying and
which is analogous to a theorem of Borda for calculating
the logarithm of a number, by means of three preceding
logarithms and a series. The same remark applies to this
class of series as to the rest : they are properly and naturally
examples of the application of the inverse method of differ-
ences, however they may have been originally obtained, and
it may not be amiss to shew how any equation such as (a)
63
expressing a relation between three values of a particular
function may become the origin of a similar series.
(68). To sum the series
tm-'(—^ ^ +2.tan-*(^ ' ^ +
+ 4 . tan - Y— ^r-i— — ^&c.
. The(4:+])«»termis
2*. tan-* 5 \ =Wx + i.
(.4(2' «)» + S.Q,'n$ '^'
Now in (a\ let 2'n be written for «, and the whole mul-
tiplied by 2% and we get
2'.tan-i5 ! 1=:
= 2' + »tan-*--l 2'tan-'-L.
2' + 'n 2'fi
The second member of this equation is etidently the
complete di£Ference of 2' tan *^ * -.j- , so that integrating both
members,
2w, + . = C + 2'.tan-»J-
and if the constant be determined as usual we find
S, = 2'.tan-'-i tan— i, S = i-tan-'i.
2'n n ft n
(69). Let /•(«) = «, /(«) rr 4 «» + S »,
/*(») = 4/(w)' + 3 ./(«), &c.
64
Required the sum of the series
tan-*— :J tan-*— -r— s + tan-*-— : — - — &c.
2/»(«) 2/c«) 2y^(n)
The (x + lf term, or u, + , is
1
w,+ , =(- ir.tan-'
2/'(«)
Now if in the equation (a) we substitute y*' (n) for « and
multiply the whole by ( — 1)' we obtain
= 2(- l)'.tan-» ^
2/'(n)
The first member of this equation is the exact difference
of
(— l)' + »,tan
because the two terms of which it consists are the successive '
values of this function due to the variation of x, with contrary
dgnsj and that without any regard to the form of the func-
tion/' (») considered as a function of n ; the second member
is equal to 2 u, 4. ^ ^ Hence
2w,+ .= A.(-.rr + *tan-»-JL
2«,4.. = 1_J2. tan-'— !— + C,
^ 2 • fin)
whence we obtain for the sum of the series
5, = ijtan— 1 + (- ly + 'tan-'—L-J;
I n f'(n)i
S = 4tan-'i,
65
for, whatever be the value of «, the quantities w, fin), /• («),
f^ (/i), &c. form an increasing series which diverges with
extreme rapidity. Thus, if «= 1, these successive values are
1, 7, 1393, 10812186007, &c.
If n then be any integer number, and the following
values may even be altogether disregarded, in a nume-
rical point of view. If we liave detained the reader too long
on this point, its close connexion with the quadrature of the
circle, will induce him to pardon the digression. We will
now resume the subject.
(70). The series 1, 5, 17, 53, l6l, 485, &c. is a recur-
ring one. — What are -r- its scale of relation, -^-its general
term, and its sum to x terms ? -
The scale is . . .......;*
• * , * 4
■ -»*•». \ ' • •
The general term is..
The sum is. 3'-^^:— l. .
(71). Which of the two series
1, 0, 3, 2, 6, 11, 23, 49, 223, &c.
1,0, 3, 2, 5, 10^ 24, 51> 247, &c.
is a r^utriDg one, ^nd what is its scale of relation ? '
. /
(72). To shew that
sin Oy sin 2 6, sin 3 6, sin x 0,
and cos 0, cos 2 6, cos 3 6...... .cos x 0,
- ■ * '
form two recurring series, and to find their scales of relation.
The reader will remember in order to prove this,, that the
character 6f a recurring series consists in the possibility of
expressing any term by one or more of Ithe preceding termsv
multiplied by invariable quantities.
* 1
66
SECTION VIL
Problems and Theorems relating to the developement
of exponential Functions^ and the properties of
the numbers comprised in the form A** o*.
The equation
A»f£, = (/* - i)»u,
discovered by Lagrange (See Appendix, Art. 358.) and the
yet more general theorem of Arbogast demonstrated in the
following articles render it desirable to possess some general
formula, to facilitate the developement of tliese and similar
expressions. We have already seen some .of the uses to
which the numbers comprised in the form A'^o* are appli-
cable in the theory of series. In what follows we shall lay
before the reader a connected view of their piroperties, which
bear cKrectly upon the point in question, and aSbrd an easy
and general solution of the difficulty. But their application
is by no means confined to this, and before we quit this sub*
ject we shaU point out their use in one or two other instances-
where they may be introduced with advantage.
(F). Prdblem. To develope f(/) any function what*
ever of /, in a series of the powers of f , or to determine A^,
jif, A^y &Cj in the following equation,
/(O = ^ + A,t + ^.<»+ ^^f* + &c.
Let /(I), /'(I), /"(l)i &c. denote the values assumed by
/(«), ^ -• 9 -4^ I &c. the several differential coefficients
67
or derived functions of f(x) when x becomes unity. Then
by Taylor's theorem we shall have (A being any quantity)
, /(l + A)=/(I)+-^^A+-^A« + &c.
Since this is true whatever be the value of h, suppose it
equal to c'— 1, and the above equation becomes
/(0=/(I) +-Ql}(e'- 1) +-Q^(/—l)* + &c.
JL 1 • ^
The coefficient of f therefore in the first member f(ef)
is e(|ual to the sum of its several coefficients in the terms of
the second member. Now, the coefficient of f in /{\) is
/U) X 0* being /(I) when Xszo and zero in all other cases.
In *^_jL:l(e' — 1) it is-^— ^^ . m
1 1 1.2 X
SS ————— , _— —— ^— ,
1 1 X
^ 4^^'' - ')*» ""-^T^^^ - 2 ^ + 1) it 18
^ •% 1.2
/'(O ^'-^^.y-hCf _ f'(\) A«<^
1.2 1 X "" 1 .ijj ' 1 i
and so on ; (as is evident if we consider that the deyelope*
ment of ^ in general is
n . . w* ., . n'
I + '^ e + /«+ — " — t' + &c.)
1 \.^ I.. ..X
Let these be collected together, and we find for the value
of Jg or the coefficient in /{e^)
1.2 xt-^ ^ I 1 .2 5
Now let the symbols of operation be separated from those
of quantity, and we get
68
^= -4—5/(1) ^ISHa +£ii>A. + 8CC. V
/(I + A )o'
1 •«•> » » X
y (I -f A) 0* being understood (as in all similar cases) to have
no other meaning than its developementj of which it is a
mere abbreviated expression, each power of A being under*
stood to be separately affixed as if hy multiplication to the 0*"
which follows. Hence this general
THEOREM.
fiff) ^fiy) + j/(i + A)« + Jl/(i +A)tf« + &c.
which will be found to comprise all the properties of the
numbers A"* o^ we shall have occasion to employ.
(2). Corollary. Hence, if by any of the usual methods
the developement oi f{e') be obtained, or the value of A,
assigned, that of /(I +A) ^ may be obtained in functions of
X and vice versa, for we have
V A • • • • rP
and,
/(1+A)e?' = 1 .2.. ..i:.^.
For example, we shall have,
(3). (1+A)«>'=1% (1+A)»^=:2%..-..(1+A)*a'= w',
whatever value we assign to «, whether positive, negative,
V integral, fractional, or even imaginary. Por,
/ ■ let /(I + A) = (1 + A)«, then f(/) = ^% or since
e-*^ 1 +?/ + ~t'+ &c.
* J . . . .jr
69
we have
m
the value oi f{\ + A) o' to be found.
Suppose for instance «= — 1, and we have
L+ A "^ '^
Now the first member of this (by the definition) has no
other meaning than
{ 1 ~ A + A« - &c. { tf*
or, ^' - Ac' + A*c>' - &c.
But in this (as in all other such series) we may omit all the
terms after A' o', they being each separately zeroy by the property
of these numbers^ (Appendix, Art* 350 ) so that we get,
0'- Ai?* + A'o' ± A^c>'='C-. 1)',
or, merely reversing the order of writing it,
A'e?' — A'-V+ ± At?' q: c' = 1,
whatever be the value of x. It may not be amiss to verify
this result by a numerical example, suppose for instance :rs=5,
and taking the values of A^^', A*e?*, &c. from the Table
(Ex. 33. Sect. 1.) we have
120—240+ I'SO— 30+1 = 1.
(4.) Again, we may prove in like manner, that
{log(l + A) }"^'«0,
unless « = ar, in which case its value is 1 . 2 .... /i. For since
(log erfszf'y the coefficient of /' in the developement of this
function (regarded as a function of e*) is zero unless j: = /?,
in which case we have A, or ^4;,= 1, and 1 . (2. . . .« yJn^
1 <^ n
70
{5), Thus taking n=i\, we have
1 2 * x
= 0,
for every value of x greater than unity, which may in like
manner be verified by numerical substitution.
(6). {log(l + ^)}V/(^)(^ = X(T- 1)
(x- n+ \)f{A)o'-\
The coefficient of f in the developement of ^" ./(^— 1)
IS evidently the same with that of /'— * in that of /(^— 1).
The former Coefficient is
Jlogri + A)}^f(A)o'
9
and the latter, .--^ ,
which being equated, the proposition is apparent.
(7). A-.-=: ^'^""^ 5'^gO^-^)|V4->,
An immediate consequence of the foregoing, changing
only X into J7 + /i, and making /(A) = A— *. This equation
enables us to continue the series,
A^(?', Ao', A-o', See.
backwards, to any extent, according to one uniform law,
though it must rather be regarded as a definition of A--"^*
than in any other light, since the value of that expression (or
its equivalent 2" cf) is not fixed by assigning only the superior
limit (0) of the integral.
(8). Prop. To shew that whatever be the value of «,
/ {(1+A)»|d'=«'./(1+A)(?'.
71
Take the identical equation
The coefficient of t' in the first member of this equation
(regarded as a function of «*) is by (1. Sect. 7.)
A • ^ • • • • Ju
but, in the second it is evidently equal to n' x into that of
the same power of / in / (^), or to
^ .
/(I + A)^
• V
Equating these the transformation in question results,
which is often of great use in eluding very troublesome deve-
lopements. Thus for instance.
{ 1 +(l + A)» I -©'=«' . (2 + Ar 0'.
(10). To prove the following very general properties of
the numbers comprised in the form A" «'*»
{/(I +A) +/(-j-^ ]o"— =»;.... («),
whatever be the form of the function denoted by/.
Suppose
«
then will f(e-') = ^o - ^i ' + ^t ^ - &c.
72
Hence, it is evident that their sum/(^)+/(^""0 contains
no odd powers of jt, and their diflFerence / (O — /(^""O ^^o
even ones. The coefficients therefore of ^"'~ * in the develope-
ment of the former, and of f in that of the latter expression
are respectively zero. Now these expressions, put under
the form / (e') ±.fl-'j and regarded as functions of ^ give,
by applying the general theorem (Ex. 1. Sect. 7.)> for the
aforesaid coefficients, the first members respectively of the
equations {a) and {b\ whence the truth of the proposition is
apparent. It may also be derived from (8) by making
fi=: — 1. Many particular cases of these theorems assume a
very remarkable form, thus :
(11). If we take /(I +A) ss ^ ,
we have by (b)
A
2 H- A
or
©•'=:0,
A^*' A'©** A't?" A"©**
= 0,
►ftr
(12). If we suppose /(I + A) = { 1 - (1 -f A) J ", we
above theorem gives the two equations -
and
Vl + A/
(--D* A»(?«',
73
both which may be mcluded in one^ by writing it as follows :
(^— A-Vo* = (-1)* + ' A^^T*.
(13). Let us take a transcendental form of/, and suppose
/(l + A)=y^log(l + A),
then we have it demonstrated (in Note iV, p. 683 to Lacroix,
Engl. Transl.) that
/(l + A)+/(j-^)«ilog(l + A)%
but by (4) it appears that log(l + Ay«^=sO, unless 2x = 2,
or xss l| therefore (this case excepted)
which substituted in (b) of (10) gives
/(l + A)o«'=0.
Now the form of /(I -f a) in this instance being the
transcendent^
^d_A , ., . v A A*
A
our equation becomes
/■
log(l + A) = -- — +&C.
Ao^ A*©** A^'o^ ^
P 2» (2«)»
which will be found verified in every case (j^sl excepted)
by actual substitution of the numerical values given in the
table. To such an extent may the separation of the symbols
of operation from those of quantity be carried, without the
possibility of error or misconception. What value the first
member of the above equation assumes^ when the exponent
of is odd, will be seen hereafter (22* Sect. 8.)
*K
74
#
(14). Theo*iiem. Let / (A) and/^(A) be any two
functions of ^, then will the following equation hold good
wher^ in the second membery the unaccented A is to be
referred to the unaccented powers of 0, and the accented to
the accented powers.
Let
/ (^^ 1) = ^, + ^1 * + ^, *• + &c.
/, (^- 1) = a^ + ^j/ + AT, ^ -I- &c.
then will
The coefEcient of t* in the first member of this, by the
general theorem in (1. Sect. 7.) is represented by
{ /,(1 -f A - 1) X/,(1 + A - 1) { g- ^
1 X
{/(A)x/.(A) \o' ^
1 X
while in the second it is
But, because ^, and a, are the coeflScients of f in the
respective devekq>ement6 /, (^— 1) andjf; (/—I) we have by
the same general theorem
hence we fiiid, ^,_, = /s^^^*^"' &c, •
I (x— 1)
75
and by substitution^ the expre^ion {b) becomes
-2 J/.(A)o'./.(A)«» + f/(A)e— ./.(A)a' +
X » m » * tlb ^ M
Now let the symbols of operation be separated from those
of quantity, keeping the powers of distinct from^ each Other,
by the system of accentuation explained in Appendix, Art.
355^ and it becomes
which compared with the expression (a) renders the pro-
position evident.
(15). By a process precisely similar, we may prove in
general that
{/. (A) x/,(A) x/3(A) X &c. \ 0'^
/,(A) ./,(AO.&c. {^-f(?' + o"+&c. }'.
(16). Hence also we may shew that whatever be th?
value of n^
\ (1 H- A)«/(A) } (,' =/(A) { « +^ r,^
(«+©)' being developed in powers oi o as a mere algebraic
symbol, andy*( A) beiiig then applied to each separate power
so produced. For, if in the foregoing proposition (14) we
write (1 + Ay* for/ (a) and/(A) for/ (A) we see that
<i + A)y(A)a' =
(l + A)V./(A)(?0-i- f (l + A)"^'-*./(A)o*+&C.
Now by 3. Sect. 7.) it appears that
( I + A)" o' = u% (1 + Z^y* o' -/ = n' - % &c.
» V
76
so that the second member becomes
rf ./(^) »• + - «— ' ./(A) «' + 8tc.
which separating the symbols of operation from those of
quantity, takes the form
/(^) J «• fl» + I »* - • 0' + i^illi^ «' — e« + &c. I
=/(A) \n-\-o\:
(17). As a particular instance of the application of this
lety( A)= a" and let » = — 1, and we shall get
A-(- 1 ^Oy =-^—<fy
or since A'-C-l +<?)' = (— ly. A"* (1 ~(^)',
m
=:(-.l)%{ A"'(>'-.A--^'(>'+ ± A'o'}-
This is the transformation we have already had occasion
to employ in (25. Sect. 1.), and that made use of in (27.
Sect. 1-) may be derived precisely in the same manner ; for
if instead of putting «= -^ 1, in the theorem in (14), we put
w= — a, still supposing f{^)= A**, it becomes
m
whence,
A-(^ a ^e?y= ^ -0%
(1 + A)
A"* (a — or= (-. 1)' . ^^ o\
(1 + A)^
and the use of these transformations in simplifying pretty
complicated expressions, and reducing them to a manageable
and even elegant form, is in the instances alluded to (and
expecially the latter) by no means contemptible. If we make
77
»i«sO, the tenns of A^ia—oY after the first (^'o' .tP= a' .
^'0"= a*) all vanish} and we have simply
«' = (-!)'. L_^<,'
(1 + A)"
as we there asserted. Other uses of the transformations in
this and the last number will shortly appear.
(18). Theokem. /(A){(a+(>)'.(6 + «?)y.(<:+o)'-&c.} =
=/( A) . (1 + A')* . (1 + A")' . &c.
The a's and their powers being referred by the accents
over them to the powers of 0^ affected with the same number
of accents.
To demonstrate it, we have
/{A){a+oy*.{b+oy=
a'/(A)(* + oy+ jK-'/(A)e(*+«)'' + &c.
= a*. { 6'/(A)e» + 1^— /(A)e' +
+ f a'-'^6»/(A)o' +|A»'-'/(A)e'+&c.l
i.a .
Now we have, 6»=(l + A)*o*} ft* — ' = (1 + A)*5»~' j
&c. by (S. Sect. 7.) and substituting these values in the above
expression it becomes
a* |(H-A)V.y(A)e''+^(H-A)V-'./(A)o" +&C.I
78
1 .2
In this we may now separate the sjfmbols of operation
from those of quantity, by employing the system of accen-
tuation, and we shall have for the value of our expression,
Jr -a'-^/(A).(l + ^'j\o"^.o'^t(i'y--^o^Jr &c.| +&c.
The series within the brackets have for their abbreviations
respectively, o^ {o + d'y^ o' {o -f o'J^ o" (;p + «?'% &c. and
writing these in their places, and at the same time replacing
fl', fl'""*, &c. by their values (given by 3. Sect. 7.).
fl' = (H-AO''e"; fl'V = (l + AO*o''""', &c.
our formula once more transformed will be
+ f •/(A)(J+^")Mo'(^ + ^VJ.(H-A')'^'-* + &c.
and again, finally separating the symbols of operation from
those of quantity^ it will become
/.(A)(l + A>(l + A''y|{^ + <?''y(e?''^'»+|^''-*()*-i-&c.)|
=/(A)(l H- A')'(l +Z^7 { (pA-(/r.{oUy \ ,
and by a similar train of operations the theorem in question
mayv be proved, to the full extent of its enunciation.
/
79
SECTION vni.
Application of the foregoing Theorems to the deve-
lopenient of particular Functions^ the Summation
of Series, ^c.
(1). 1 o develope in powers of t.
/(^) = TT^ = ^0 4- il» * + A.f' + &c.
1 +r
•'^ 1+(1 + A) 2 +
o'.
Therefore, by Ex. 2. Sect. 7.
1 .2 .... X 2+ A
_ 1 Co* Ao' AV^7
thus we find ^0=1--= i, ^.=:iifo-i)=-i,
^4^=0, and so on, so that
^ = 1 - 1 + <' . &c.
1+^' 2 4
This is the great advanta^ resulting from the employment
of these functions : any series of them such as
«.(?'+ 6 iCi d'+ r A"o'+ &c.
however complicated, necessarily breaks off in a limited
number (:r+i) of terms, and thus enables us tb assign in a
CG^paratively simple form, the general terms of an unlimited
80
variety of developements, jvhich would otherwise be scarcely
expressible without having recourse to the combinatory an-
alysis, which ought never, in my opinion, to be employed, till
every artifice of abbreviation, and every refinement of analysis
has been found unavailing.
(2). To devolope (^ — l)* in powers of t,
/(0 = (^-ir; /(l + A)o'=(l+A~l)*^'= A-o';
{e'-\Y^A^(P+ ^^ t + ^=-^^'+ &c.
11.2
n JH A » ^M 4- 1
=:-A-i_>/*+ _^_? r+»+&c.
1 .... « J .2.. .(w + 1)
This elegant expression was originally given by Mr. Ivory
(Leyboume's Repository, 1804. Quest. 60.) and afterwards
by Dr. Brinkley, Phil. Trans. 1807- i. Both these Geo-
meters arrive at it, however, by a different method from that
above pursued. The former mentions it only incidentally,
nor does the latter, who pursued the subject much farther,
seem to have perceived the system of which it, and several
more of the truly beautiful theorems he has given in that
paper, form a part. To him, however, belongs the merit of
introducing the numbers comprised in the form A'"^** among
the data qf analysis, as objects of ultiniate reference. I ought
too, to notice that the developement of ^ in the form
above given (1. Sect. 8.) is also to be found for the first time'
in his paper.
(3). To develope in powers of /,
e* — 1
81
and consequently the coefficient of t*, or
1 .^....xti 2 3 •• • " a + lV
This expression for the coefficient of f is given in the
Phil. Trans. 1815. in a paper, ** On the developement of
Exponential Functions."
It has been already shewn (Appendix, Art. 408.) that the
odd values of ji, in the developement of -^ (^^ excepted)
all vanish, hence we see that the following equation must
hold good for every value of x except unity,
+ + = 0.
2 3 2x
From the same article, it also appears that the coefficient
of t^ in the same developement is
V *^ -1.4. ...2^'
which compared with*^^^ — , gives the following very
i*i2*»«*Sbkr
simple expression for the numbers of Bemouilli,
A
0^
or, B
aj — l
Thus we may calculate £he numerical values of these
numbers, with a degree of facility far surpassing that afforded
by the expression demonstrated in that article, for instance
we have
2^3 6' ' 2 34 5 30 '
and so on.
m
(4). To develope ( — j in powers oF U
This function being equal to ^-7^-^ ) > the coefficient
of f will be by (Ex. i . Sect. 7.)
1.2 X (.A3
Fo^ the developement of this (unction int6 « fbrmula
adapted to numerical computation, the reader is referred to
the paper " on the Devetoptment of Exponential Functions **
above cited. Or he will find the original function completely
developed in a hiost elegant series, in Dr. BrinUey's paper
above noticed. Our object is to remark that since by (Ex. 7*
Sect. 7.) we have
1.. . .(4? -f w) c A 3 1 ^
therefore, the coefficient of /* ^ * in the proposed function,
or, which is the same thing, that 6f tf in
(^-1)-"
is properly represented by , and therefore that
J. • i? • • •' * X
the equation
(^ _ ,). = ^^:£_*» + __^lp_ r- + &c.
1 . . . .« 1. . . . (n + 1)
proved by Dr. Briukley to subsist for positive values of /i,
k now shewn to hdd good also for negative.
J .
(5). To develope e' in powers of f .
J(/)^^\ /{I + A)o' ^e^-^^tf ^e.e^o%
whence we deduce
S3
1 • Z . . . . iZ*
= i" "^ "^ + . . . •- \.
Thus A^ = e, A^ss e, ^,= , A.9b , &c,
• 1.2 ' 1.2.3
and
^ =^ + ^- + 2^. — — + 5 ^ . + &c.
J 1.2 1 .2.3
(6)1 To develope —^ and in general {a + O" in
powers of /,
ss — i- <0 T — . » + — • -r — - v ' ' • • • •
1.2....H 1 !+« 1.2 (1+a)'
. y(w-I)....(»— J?+0 '^o'
••• ■' 1.2....* (l+fl)*'
and in the case proposed where assl and »= - -> the value
of -4, is
\ !(?'- 1a^+ — A«^'-
.1.2 xl 4 4.. 8
v/2
. 1.3 (2a:— 1) ., .
4.8.. ..(4«)
and in the sanie manner may any algebraic function of ^ be
developed with little trouble.
84
(7). To extend the above mode of developing /(/) to
exponential functions of tviro or more variables of the form
/(^, ^, ^% &c.)
Let all but / be regarded as so many constants^ entering
into the composition of the given function, regarded as a
function of ^. Then will the coefficient of any power of U
as f be, by what has been proved,
" I ■
1 • <^ . • • • uG
Into this function / does not enter, but, being a function
of ^', &c. it is itself developable in a series of power of
/', t'\ &c. Let the coefficient of /'^ in this developement be
sought by a similar process. It will be
/{ 1 -fA, 1 + £^y ir\ &c. \(f.o'^
1.2,., .J?xl .2. . . .y
the powers of A produced by this second process being kept
distinct from those resulting from the first . and applied to
their proper power o^ of o, by the accents affixed to them ^s
in (Ex. 14. Sect. 7.)- THis then will be the coefficient of
f*t'^t in the developement of the proposed function, regarded
as a function only of / and t\ Proceeding in this manner, and
denoting by -4,, ,,,,&«. ^^ coefficient of the combination
/« . f'9 , ^"« . gf^c. in the final result, we have
J _ f\ 1 -HA, 1 +A^, 1 +A^^, 8cc. } o'.o^.o''>^c.
',y,r.&c. J 2 XX 1.2 J/Xl .2 zx^c.
(8). To develope /{ ^■*- '' + <" + &c. j in powers of t, t^,
8tc..
In this case
- _ /{( + A)(l+A^)(l+A^').8cc. \o\o'.o'\^c^
^x.y.«,&c.— ^2 xxl.2 J/Xl.2 zx&c
Now let i:(l+A)*.(l+AO'*.&c. be any term of the
developement of / { (1 + A) (1 + A') • &c. ] . Then when
8^
this is prefixed to (f .0'^ , 0"^ . &c. in the manner denoted hj
the accents, it becomes
iC . (1 + A)» 0* . ( 1 + A)* i?^' . (1 + A)» (»' . &c.
= K . «» . «y . w' . &c. (by S. Sect. 7.)
Hence it appears that the same series of terms will result
from
/{a + A)(l + A').&c. \ i^.o'y.Scc.
as would have arisen from
/(l + A)<7' + ^+'+*%
and consequently, that
^ _ /(l + A)o'-*-y-^'+^
*,y,*,&c. ^ ^ aj-x 1.2 2/Xl .2 jz; x &c. '
Which is also deducible from the theorem for raising a
polynomial to any power, combined with that in (Ex, 1.
Sect. 7.) and xiice vers&y the multinomial theorem is directly
deducible from this.
Hence too it appears, that the developement of
y(^ + // + &c.) jg directly deducible from that of /(^j the
coefficient of f .P . f* . &c. in the former being equal to
that of ^ + '^ + ' + **^ in the latter, multiplied into
1.2 (j:-f2/ + ;2: + &c.)
1 .. . .a'x 1 . . . .yx I .. . .z + Scc.
(9). To prove that
1 o*' 1
2+ A 2x
JB^— I being the x* number of Bemouilli.
By (Ex. 1. Sect. 8.) —-^ — o^'-» is the coefficient of
^ 2 + A
86
i*"-' in the devolopement of , multiplied by 1 . 2 . S . . .
(2 a?— I), that is, to the coefficient of i^ in , multiplied
1 +^
by the same quantity. Again, by Appendix, Art. 408. it
appears that this last coefficient is equal to — (2*'-- I) x into
the coefficient of the same power in -^^^ , which coefficient
^ — 1
is ( — ly + » , iLZJt — • Hence we must have
2 + A
1 . 2.... (2 j:-1)x -(-])' + ' ^'^ ^ .5,,-.
— (" I) ' •■t^ts — I*
24r
(10). To sum the series
l»-2» + 3''-4* + &c. flrfiw/;
Since by (Ex. 3. Sect. 7.) we have (1 + A)o*sz 1-,
(1 + A)*«* = 2*, &c. therefore the prpposed series becomes
by substitution
S = (1 + A)c^-- (1 + Zi)V*+ (1 + A)'fi*- &c.
or, separating the symbols of operation from those of quan-
tity
S= }(1+ A) -(l + A)*+(l+A)»-&c. |()"
1+(1 + A) C 2+A3
because this function developed in powers of a will neces-
sarily produce the same series as the other. To throw this
87
into a calculable form, let it be actually developed in thi$
manneri and we find
Thus we have
1—1 + 1-1 +&c.= i,
1—2 + 3-4 +&c.=:i,
4
1* - fi' + 3* - 4' + &c. = 0,
l** - 2' +3»- 4' +&C. =1,
o
|4 _ £4 ^. 34 _ 44 4. ^^. -- ; Sic.
All the even values vanish, and it ought to be so, for the
substitution of 1 for / (1' + A) in (10.
2 J 1 +(1 + A) I
Sect. 7.) equation (ft), gives
1+(1 + A) 2
in every case except when xasO, when it becomes - . The
odd values may be expressed by the numbers of Bemouilli>
for writing 2x—l for /i, the general e3q)ression for the odd
values of S becomes
-—!-(»—' = (-!)' + «. 1_-LjB^^.
2+A 2x ^ '
by (Ek. 9. Sect. 8.): a result exactly agreeing with that
deduced by Euler in his Instkutumes Calctdi differenttaHs^
Cap. vii. p. 501. from a principle, it must be confessed, not
at all satisfactory.
(11). The series
l** + 2" + 3"+ 01^= S,
88
being proposed^ if we treat it in the same way, we get
«,= {(1 + A) + (l + A)«+.-..(l + A)'}d«
A
which, developed in powers of a, gives
1.2 1.2.3
1.2 («+l)
This is different in its form, from any expression we
have yet given for the sum of this series. It may, however,
be obtained by resolving (flf + l)" into preceding, instead of
succeeding values» and integrating, as in (Ex. 26. Sect. 2. and
Ex. 23. Sect. 6.)
(12). To sum the series of (10) viz.
l« - 2- + 3" - &c.
to X terms.
Here 5,= { (I + A)- (l + A)»+ (1 + A)»
±(l + A)']o*
_ ( l + A)- (— 1)' (1 + A)' + »
2-f A
•
This expression, developed in powers of a aflbrds a cal-
culable value of Sj,, the number of whose terms can never
exceed n+ 1, and consequently the developement need neVer
be carried farther. Thus we have, for instance,
1 - 2 + 3 -
= c-,,...(i^).i.
89
(13). To sum the series
r.* -!-2".<' + 3*./» + &c.
to X terms and to infinity. This series treated exactly in the
same manner, becomes
5,= { /(I + A) -f <«(1 + i^)'»+ . . . .<'(1 + A)' } (»«
*(1 + A)-]
Let S represent the series to infinity, then
{\-t)—tA ll-t (l-0(l-<-< A)>
= T^{'-"+(i^«)^'+(n:7)''^ ••■■*■
(l^,)"--}
Thus we have as particular instances
1 .^+2.<* + 3./'+&c.= t: — T»>
(i~/y
(l-O' (1-0'
The expression for S, may, in like manner, be easily
developed in powers of A, and will then assume a calculable
form, but we prefer leaving it in its present state. The
reader may, if he please, supply this part of the operation.
90
«
(14). In general to sum the series
having given
Jo -f-^i . ^ + ^8 . ^' + &c. = i^(/).
The series treated as before gives
S= J Jo + ^,(H-A) + &C. }i?«
= F(l-fA)o«
= iJ'(i).(?~+£-i-: A(^«+ &c.
\ 1
when adapted to actual computation by the developement of
Fd+A) in powers of A .
(15). To sum the series
S ^ P+. ^^ + ^-1^ H- ±-'J^ + &c.
1 1.2 1.2.3
This series being the same with
^"^^ ^-t — Ah- t — A»+&c.
1 1.2
by substituting ^for F(t) and »+ 1 for « in the last problem
we find
C 1 1.2
1 («-hl)
Thus, if we suppose A= 1, we find
■}
2 3
\ 1.2. '
91
11.2 '
P-f - -H h &c.= 15 e. and so on.
11.2 '
(16). r- 3«+ 5**- &c.= 5'.
Then,
1 -|.(l + A)* 2(1 +A) + A«
The developement of this in powers of A would be attended
with some trouble, but this is not ihdispensable. It may be
reduced to a limited number of calculable terms in many
different ways, the simplest of which seems to be as follows :
S= \ 1-l-A ^(l-|-A)A-_^(l-f-A)A^^g^^7 ,^
C2(l + A) 4(l-t-A)> 8(1-|-A)» >
2 4 1-t-A 8 (l + A)*
the number of whose terms can never exceed - -t- 1 and each
2
term of which is easily calculated. In fact, whenever n is
an odd number S vanishes, for, if in the equation (a) of
(10. Sect. 7.) we suppose
/<'-^^^=rRrrAy'
there will result
1-1- A
l-f(l + A)«
o*'-» = 0.
and when n is even (writing 2 .i f or it) the value of S
becomes
o"" 1 A* ,,^ _^ 1 A*' »,
2 4 1-t-A 2' + » (l-fA)'
Thus we find
1 — 1 + 1 -. 1 4- &c.= i
C*
1 - 3 + 5 - 7 + &c.= 0,
1*- 3«-f 5'- ?•+ &c.= ~i
P- S'-f- 5'- 7'+ &c.= 0, &c.
See also Note i). Lacroix, TransL p. 360.
(17). To complete the theory of th.ese sums we shall
subjoin the following example^ which will call -for the em-
ployment of nearly all the transformations above demon-
strated, and will thus^ by illustrating their use and manage-
ment, prove the more acceptable as the mode of investigation
followed in this and the last section is, if we mistake not,
perfectly novel in analysis.
To sum the series
+ — &c. ad inf,
tiVt -4-1 ./
and to express its sum by means of the numbers of Bernoulli.
Let C„ 4. , represent the sum of the series, or
C\=i- 1 + i-&c.
1 3 5
C, = 1 - i + i - &c.
' P 33 5*
Then Euler has shewn that, ^ being any arc, the develope-
ment of sec % will be
2* ^ 2* ^ 2*
sec^ = ~.C, +^€3.^* -f =-C;.^ + &c.
^) X -the coefficient
93
of a** in the developcment of sec 6. Call this coefficient J^,
then since
sec V = » .,, =: ■ ,
COS ^ ed%/-> 4- ^--«%/-> '
we have by (1. Sect. 7.)
1 2
A^= X - - 0*^
1.2....(2j:) (I + A)v'-» + (1 + A)-v'-' '
but since by (8. Sect. 7.)) writing 2x for x and >/ — 1 for «
/ { (1 + A) v^- • } »" = (v^- 1)" ./(I + A)o*',
we have
•* 1.2....(2x) (l + A) H-(l + A)-" '
which'substituted in the value of Ca,+ , gives
^ _ 2/ 1-t-A „
'"^-»-*'" 1.2. ...(2a?) •(H-A)* + l'' •
The latter factor of this expression coincides precisely
with that obtained in the last number for the series
!•* _- s^' + 5" — &c.
and its numerical value may readily be computed by the for-
mula there given. Hence this remarkable relation between
the two series
1.2.3 (2 *) X < ——^ ~ -— TT -f — -: &c. J =
-) X } 1*'- 3«'+5«'-.&c. {
and as particular instances,
+ T - &c. = -.
13 5 4
-r — — ; + — — OCC. 3= — , &c.
P 3' 5» 32*
94
The transformation into numbers of Bernouilli may be
accomplished as follows.
l+'A (1 + A)*
(1 + Ay+ 1 (1 +A)|(1 + A)»+ J {
~ 2(1 + A)l (I + A)« + li
I 1 (l+A)_(l+A)-«
+ -z •
2(1 + A) « (H-A)« + l
Now, by (3. Sect. 7.) it appears that 0**= 1, and we
therefore have
^ — ■■ I !r n: ri :: v •
(l+Ay-fl 2 2 (l + A)«-fl
Now, in the theorem (16. Sect. 7.) if we make y (a)=s
1 ^ i 1 .
, and « s= - and in succession, it gives
2-fA 2 2 ' ^
2 + A 2+aV 2/
2-t-A 2 + aV 2/
so that by substitution we get
(1+A)'+1 2 2 + AC\ 2/ V 2/>
^1 a^—Cg. 2 a(2a-l)(2^-2) 3 7
2^2 + aIi ^ 1.2.3x2* -r«xt..y
95
Again, we have, by (9. Sect. 8.)
^ .^-» = (-1)'.?1zJLb„^.
2 + A ' 2x
whence,
1 • o** — «— 1
_L^o»'-»=-(-l)'.r^ -liJ„_,;&c.
and consequently^ by substituting these values, our expression
is cleared of the symbols A and o, and reduces itself to
and hence we have, finally^
(i) ,
1.2 (2x)l
;«jr -f- 1
V 1 ** * 1.2 3x2» «'"-3^
2jr 3 2*~-l » "^ ^
^" '* 1.2 (2a: - 2)'(2t- 1) .2''-' V 3'
The practicability of expressing this function by means of
the numbers of Bernouilli was first shewn, and the above ex-
pression demonstrated in the Phil. Trans. 1814. The demon-
stration there given is however very circuitous and rather
obscure. The above has the advantage of connecting what
was before an insulated result with the general theory of
series of this sort.
(18). Given the sum of the series
or the generating function of ^„ required the sum of
96
A,.r\t + ^2(i"+2")./«+^3(rH-2»4-s»)f«+8cc.
or the generating function of
By substituting for 1", 2", . . . .a:*, their values
(l+A)o% (1 + Ay (?%.... (1+ Ay a«
we find as in (1 1. Sect. 8.)
So that our series becomes^
A A
= i-^t_^ I A,t{l + A) + ^,/*(l + A)' + &c. }(?»-
A
— ^'^^c^, { Alt +A^e + iic. I
A
£^ A
= i-i-^ {F(t + tA)- F{t) \ e'
(19). For example
L%il±!:^ilt!l±^-i.&c.==
1 1.2 1.2.3
g.(l4.A)l-Il- <?«=/< 1+ A+ 1 A2+&C.?
A C. 1,2 1.2. 3 >
= ^ lo^ +~ -5-Aa» H- — i A«^* +
C. 1 .« , ^ ^
<)*
1 .2.3
^±? A''.'^}
1.2....(« + 1) >•
97
(20). I'.t + (l" + 2")<' + (l»+e"+3")<> + &c. ad inf.
A li—t-ta. 1-ty
o""
(21). It is required to prove the following expression for
the numbers of Bernoulli,
B„^. = (- 1)'+' . >^±1 — LLJL^
}•
(2 37 -f 1)
Take the equation (Append. Art. 411.)
'ZUg =/ii,d« — -i H !- .-7-M, — &c.
•^ 9, I .9 dx
in which make u^si a?^% and we shall find for the coefficient
of X in the developement of 2 (a?"')
(- iy + «B,,-,.
But, if we integrate the expression given for x^ in
(2. Sect. 2.) -we find
2(^)« C+(-ir. |f.A.» + *-^^±PA*d» + »+8cc.}
If this be developed in powers of x^ and the coefficients
of X collected together, we shall find for the whole coefficient
of that term
*
N
98
In this if we write t i for riy we shall find by cohiparing
it with the coefikient previously found
^2.
-, = (-.)-{^'-il^^*c.}
which is, in facf, the equation to be proVed. If 9,1 — 1 be
put for n the term x vanishes form the integral 2 (a?"),
(See Appendix Art. 36^.) which shews that M^e must have
—A -t- &c. = 0,
which has been already proved by a different process (13.
Sect. 7.) It ^in not be amiss now to notice one mofte pro-
perty of the numbers comprised in the fofih A*" ^ by the sUd
of which the table of their numerical values given in (33.
Sect I.) may be continued to any extent, with very little
trouble.
(22). The numbers
A"<?% A«<?» + S A* <?'* + % &C.
form a recurring series. To shew this, and to determine its
scale of relation.
Since (App. Art. 350.)
A" o"^ = « '— " (« - 1)'+ ± - . 1%
K 1
this function is of the form
It is, therefore, a particular integral of some equation of
cbKiiRice» of the first degree with constant coefficients (App.
Art. Sr89.) and is therefore (App. Art. 390.) the general term
of some recurring series. Let now
A= 1>
V
j:lf,= 1.2+1. 3 + 2.$+ .. ..(«— 1).^
urf«= 1 .2.3, . . ,«.
Then (App. Art. 389) This equation will be
A»*^ + »- ^. A* <»'■*"**"'* + .. .. i^nA^^c?
^a that the scale of relation is
SECTION IX.
Exercises, 8gc. in the Interpolation of Series.
(1). In any series of consecutive e<}uidistant values of
a function, where one is deficient, to insert that one.
Let the equidistant valuer i>e
and let the deficient one be v^ so that all but Vf ar^ given.
A^^ume A^^'o ^ Of or that the (»— 1)* differences are con-
stant, which will almost always be nearly tk^ case in tabulated
results, except under extreme circumstances. Then we
have
A» w . «(»— 1) ^ n ^
A*Vo= Vn— - v»— 1+ ; ^ t;^-.a ± -v.:?: Vo= 0,
1 i • 2 J
an equation of the first degree, from which any one of the
values as i^^may be determined in terms of ti^e rest.
(2). Given two values of any function, required to insert
one equidistant between them.
100
Given % and v^ required r„
A* tlo= 0, ^2— 2 V, + Vo=5 Of
Vo + V,
Vi =
(3). Given three values Voi Vj, v, of any function to
insert the deficient one v^.
Va = r • •
In like manner, if v, were the deficient value, we should
find
(4). Given the following common logarithms,
log 510 = 2.70757018
log 511 =2.70842090
log 513 =2.71011737
log 514 = 2.71096312
it is required to insert the deficient value log 512.
Given v^ = log 510, v, = log 511,^ v^ = log 515, and
V4=log 5 14. Required Va=log 512,
A*Vo= V4— 4 V3+ 6 v,+ 4 v,+ Vo=
^^ ^ *(v,+ v,)-{v,+ v,) ^ 2.70926996
precise]y{as the table."
(o). in any series of consecutive equidistant values
where two are deficient, to insert those two.
As before, let them be
Vo> V,, Vn-h 1
101
and assuming A" i/o=0 and A'* v.=0, to obtain a continuous
law of increase or diminution throughout the whole seriest
we have
1 1.2 I
« , «(«— 1) ^ ^l
two equations of the first degree which suffice to determine
any two of the values in terms of die rest. The same prin-
ciple will serve to insert any number of deficient terms.
(6). Given v^^ v,, v^, v^. Required v, and Vy
Assume A^v^ss and A^v^s 0, then
v^- 4^3+ 6v,— 4v,+ Vo= 01
V5— 4 x;/-f 6 V3— 4 v, + Vj = *
whence
_ — 3vo+ 10tvl-5^4--2v5
10
— 2 Vo+ 5 V, + 10 v^— S V5
' 10
(7). In a table of the values of fd x(log -1 taken
between Ae limits j:=0 and d?=l *, we find the following
values corresponding to the annexed values of a.
a = 1 .326, / = 0.89S8710 ;
1.328, 0.8936220;
1.329, 0.8935004;
1.331, .-...0,8932628.
* Legendre» Exercises de Cakul Integral p. 302.
log
Vt?>t^^ ihe values corresponding to a ss 1.597> ^^4
iixttfn v^i v., Vj, V5, required v^ and v^.
„, ^ ^,- I0r,-^a0^,+4t>. ^ 0.8937455
' 15
«, = ^-^s+g0^3-10%-*-^-. ^ O.89SS807.
(8). In any series qf consecutive equidistant values,
where one or more are deficient, and the rest given, to inter-
polate any intermediate value whatever.
Insert the deficient equidistant values^ ^nd then inter*
polate the series so completed by the formula,
v. = 1;, + ^ Av, + ''^'I'Z^^) A-t;, + &c.
i 1.2
For instance,
/
(9). In a table of the values of the function tan~- (x)
or arc (tan=j7)* we have given
tan-> 10=1.471127674, tan-» 11 = 1.480136439
tan-* 13 = 1.494024435, tan -^5= 1.504228163.
Required tan — * (11.63).
The values of tan""* 12 and tan — * 14, first of all inserted
are respectively 1.487655094 and 1.409488856.
Let then «= 1.63, i?o=tan — * 10, &c. and we find
v^ = 1.485022707
the number required.
(10). Given the values v^y v^, v, of a function, required
to interpolate any given value as v„,
^3 - 9 V, 4- 8 Vo , v, — 3 1?, + 2v a
* Given by Spence in his Logarithmic Transcendents, p. 63.
103
(11). Two observations of a certiiin quantity Irere tnade
« the interval of a day from each other: the firtt gay^ for
its value tiy the second — *. When was its value zero ?
Given v^ and v,, required n so that v*£sO.
I St In general t), = t?„ + 'i (v, - v J
and making this zero,
«=- ^^
V, — Vo a + 3 '
which id the tin»e in fhtctions of a day, from the first obser-^
vation to the moflHient required. Having but two observaticMds
Weisiu^fK)^ A^t^o 8tc. 2ero. Tbk is the most ordinary id^
Stance ti interpolation. To render it exact, we should, if
^ssible, choose such opportunities for observation, as wiU
allow pf our neglecting A^i?^, &c. that i«, when the variation
of the quantity observed is nearly uniform. Such is that
of the sun's declination near the equinox, of a planet's latitude
near its node, &c. The rule for proportional parts in loga-
rithmic and other tables, depends likewise on this problem.
»
(12). Three observations of a certain quantity were
taken at equal intervals. Its value were found to be Vo>
^19 ^9> between which ks vahR« was a. When did this
happen I
Given v», Vj, v^. Required n SO that t>»sis<i,
*
make
J. 1 • ^
whence
+ \/(A' V, - <^Z A v,Y - 8 (y, - a) A* v, I .
104
The positive sign is affixed to the radical, because th^
supposition A^v^^O, or A^ v^ very small ought to reduce the
value of n to that given in the last example. If this expres-
sion be developed in powers of A^v^, and its square and
superior powers neglected ,we find
in which we may regard the term
- (v, - a) . {v, - a) . \ ,
2 (A Vp^
as a correction to be applied to the value of n calculated from
the terms v^ and v^ alone on the supposition A* vq = 0, and
thus in certain cases dispense with a troublesome calculation.
We may observe that by a proper choice of the quantities
represented by %, Vi> &c. the quantity a may always be
made zero, so that we have
- ^o»'^i.' A*1?o
2(Av,)«
for the correction to be made in this case.
(IS). Given three values, not equidistant, of a function,
to interpolate any intermediate value.
Given Va> v^, v^, required v**
Suppose the indices a, /$, 7, to be equidistant values of
some other function, thus let assz^i P=^^i9 7=^3 and let
n=2:„ then will '6a ^ vfi, Vy, v„, be the values of the function
Vg^ corresponding to the values 0, l, £, and x of the index x
the independent variable. Let v,^=w*, then Wo=Va> ^1=^/8,
ti,ssv^ UgSsVnj and the formula in Art. 404«. gives
(« — /5) (« — 7) ,, , (« — «) (« — t)
(a — ^) (a - 7) (P — o) (^ - 7) ^
(7 -a) (7 -/?) ^'
105
{14). Thr^e obsiervations of a quantity near its maximum
or minimum, are made at given times (equidistant or not.)
From the observed values^ to determine when the maximum
or minimum took place.
When a quantity is near its maximum or minimum, its
values cannot be interpolated from two observations, because
such interpolation requires the supposition of uniform varia-
tion during their interval, which cannot be made in these
circumstances. In fact the function Vo+ - (v» — ^o) does Jiot
admit a maximum or a minimum by the variation of n. In
the case of three observations, however, suppose a, /?, 7, to
be the times (from a certain epoch) at which they were made,
and ^a> '^Ai ^ 9 ^^ observed values, then, since at any other
time n we have v„ = the expression in the last problem, if we
differentiate this relative to fi^ and put the result =0, we shall
find
(/^' - y')Va- (g^- 7")^^ + («^ - /^)\
' 2 { (/? - 7) Va - («~- y)^0 + (« - f^)\ i '
the value of rt required, at which v^ is a maximum. By this
formula may the meridian altitude of the Sun, or a Star, for
example, be found when an observation precisely on the
meridian cannot be had.
If the observations be equidistant, and the epoch be. fixed
at the first, we get
" (15). Given any number of values of a quantity observed
at given times, not equidistant, to determine its value and
those of its differential coefiicients at a given instant, the
time being supposed to increase uniformly.
Let the given instant be fixed on for an epoch, and call
/ the time elapsed since that epoch, / being indeterminate.
n
106
and negative for all observations preceding the epoch, also
let a, /?, 7, &c* be the values of t at the moments of observa-^
tion, and v^, v^, v^, &c. those of the quantity observed, and
we have by Append. Art. 404. for its value v at the time t.
The values then of the diflFerential coefficients^, —,
&c, when / ss 0, or at the epoch, will be the coefficients o£
/, ^*, 8cc. in the developement of this function, divided
respectively by i, \ .% 1 . 2 . S, &c, or calling them £l ,
d t
V ^ S /?.7«^»-' r ft.7.s... .o ">
The sign prefixed to the right hand members of these
equations is the upper or lower^ according as the number of
observations, or of the letters «, ^, 7> ^> • • • • is odd or even,
Laplace's method of computing the orbit of a comet,
turns upon the application of this problem. The formulae
here deduced are somewhat different in their form from
those employed in the Mecanique Celeste, and perhaps,
rather more complicated to the eye. But in actual computa-.
tion they will, I believe, be found more convenient, the terms,
of which they consist being better adapted to logarithmic
computation, and in reality less intricate in their formation,
and in consequence affording less room for mistakes on the
part of the calculator.
107
SECTION X.
Application of the Calculus* of Differences to the
determination of Curves from properties involving
consecutive points separated hy a finite interval,
(1)/ In the circle, any line ACB (Part. III. Fig. 1.)
drawn through a certain fixed point C (the center of the
circle) and meeting the curve at its two extremities, is of a
given length in all positions of the line. It is required to
determine whether any other curves possess the same pro*
perty, and if so, to include them under one general equation .
In other words : Required the class of curves whose diame-
ters are invariable.
Draw any line CMy and let the angle MCJ=^6, CA^r,
CB 5= r\ and suppose r = ^ (6) to be the polar equation of
Ae curve sought. Then will r' = (^ + t) and since by the
condition of the question r + r =:constant = 2 a, we have the
following equation for determining the form of 0,
0(a)-0(^ + tr) = 2«.
Suppose now 2; «» - » or d sz v z, and this equation
ir
becomes
This is in fact, an equation of differences ; for, if we
suppose (tt z)ssug^ we get ^ { w . (z + I) J = w, 4. , , and
No\v^ this equation deprived of its last term ^ a, is
evidently satisfied by cos fr z, because
cos ir Z + cos TT (Z+ 1) =
108
Hence the complete integral i$
Ug:s a + C . cos w z,
C being an arbitrary constant^ or rather according to the
remark in Appendix, Art. 368. an arbitrary function of
cos 2 TT 2f, or in general, any quantity which does not change,
by the substitution of 2f + 1 for z.
IJence, restoring Hie original denominations
r ^ a + cos 6 .y (cos 2 0),
where under y(cos 2 6) are comprehended all functions of 0^
whether algebraic or transcendental, which do not change
when e fir is substituted for 6. Thus if /(cos 2 0) = O,
r=sa the equation of the circle. If /(cos 2 ^)=i, we have
r =: a -h t . cos 6,
which represents a curve similar to that in the figure, whose
algebraic equation is
(2). instead of supposing the sum of the parts AC, CJB^.
(Part III. Fig. 1.) constant, let their rectangle be invariable.
Required the class of curves possessed of this property.
Retaining the same denominations, we have now
rr'^ a\ or (a) . ^ («• + ^) = «•
which treated in the same manner, by supposing Q^irx and
(d)=w, gives
which is eridently satisfied by
because cos tt (z + 1) = — ^o^ t t. This then, containing an
arbitrary constant C is the complete integral, and as before
replacing C by an arbitrary function of cos 2 tt ^, and restor*
ing the value of^;5•
r = <^(^)«rt./(cos9<')^^^^
109
Thus the oval whose equation is r = a e^^^ satisfies the
condition, but it is also satisfied by an infinite variety of alge-
braic curves, as we shall now shew.
We have already remarked that /(Cos 2 0) may be any
function which does not change by writing v + for B. Let
F (cos 2 0) be any other such function, and it is evident that
the expression
F (cos ^ ^) + cos $
F (cos 2 a) — cos 6 '
by that substitution has its numerator and denominator in-
verted, because cos (w + ^)= —cos ^, hence if this expression
be raised to any power such as cos 9, or , &c. whose
cos 6
sign only is <:hanged by the substitution, the function so pro-
duced will remain unaltered. We are at liberty then to sup-
pose (y cos 2 6) of the form
ft
C jy(cos2^ + cosg '^^^
1 i^ (cos 2 ^) — cos eS
which value being written for y cos 2 ^ in the expression of r
above found gives
_ ^ i*'(cos2^) 4- cos ^ 7 **
^ " ^ |i?(cos 2 ^) ~ COS ^ >
which always gives algebraic curves by assigning an algebraic
form to the function F. Thus, if we suppose jF (cos 2 ^)= 1
and « = , we get for the equation of the curve
2
r =z a . (tan i Bf.
If we suppose
\^hich evidently remains unaltered by the substitution of
T + ^ for ^, we get
110
which is the equation of a circle, the pole round which the
angle 6 is reckoned being any point however situated. In
fact this property is proved to belong to the circle in the
35th and 36lli propositions of the third book of £uclid.
(3). The conchoid is produced by the revolution of «
straight line round a fixed pole, one of its points being sub-
jected to move in a straight line, while the other describes
the curve. To find a curve, or class of curves, susceptible,
of being (fescribed like the conchoid, with this difierence,
however, that instead of the directrix being a straight line,
it shall be another branch of the curve itself to be found.
The straight line CM (Part III. Fig. 2.) is to revolve
about C» so that MM' cut off between the intersections
M and M\ with the two branches A M and B My shall
be constant and = a. Draw MPy M' P' perpendicular to
CP\ Let CM^r, CM'=r\ CP:^x, PM=y, CF^oif,
CM'=zy.
Therefore r* — r -=■ a^ or r' = a + r.
Assume y zs <!> \/(x^ + J/') = (^) for the equation of the
curve. Then, since the same equation is common to both
branches, we must have also i/=: <p(r').
Now, by similar triangles CPMyCP'M'i we have
Therefore
, r
4^(/)^<Pir) _ » (fl 4- r) _ » (r)
- — , or 1
r r a + r r
an equation from which the form of <p is to be determined.
Let r = az, then a + r = 11(2: -}- J) and if we suppose
f-2 — i =: u, we have
az
IJ I -
whence^ u^ =:/ cos 2 trz, and therefore
- =f(cos 9,ir .-).
Thus if We suppose / ( cos 2 w - ) =: sin 2 v T , and ob-
serve that^ = sin AfCP = sin e. we have
r ' ^
sin 6 ss sin 2 9r « . or r=a . — ,
the equation to the spiral of Archimedes, in which the next
inferior convolution of the curve, supplies the place of another
branch. In fact, the preceding analysis does not take in the
condition that M and M' should lie in different branches^ but
the following solution will apply to the strict letter of the
enunciation.
Let C ^f be regarded as a negative value of r, answering
to one half revolution more of the line CM in which case
the gepmetrical equation r ~ r = 2 a will be represented in
analytical language by (p (Jd) •¥ <p (v •{- 0) = 2 a, and thus the
equation of the problem (1) resolves this case, provided we
select only such curves as have the property described.
Thus^ in the result
r = a .+ cos ^ ./ (cos 2 6),
when $<'ftf r must be positive, and when greater, negative^
or at least, if in any part of the variation of 6 between o and
7, r becomes negative, it must be negative in a higher degree
in the corresponding part of its variation between «r and 2 ir.
Such a curve is
r = a(\'^ -J— ^3 or {x^ + !/•) (y ~ aY = «?y*
V sm 67
and an indefinite variety of algebraic curves, among which
are some which satisfy the geometrical property r+^= 2 at,
112
with one pair of branches, at the same time that r—r=i^a
is satisfied by another pair.
(4'). Required the nature of the curve A MM' (Part III.
Fig. 3.) when the line AMM' revolving round A has the
sum of the iw* powers of the segments A M, AM* constant,
or yet more generally, when one segment -4M's=r'is any
assigned function a (r) of the other AM ^ r.
As before, suppose y =0 (r), then we must have y'^^[r\
Now since t'^iaiy) and by similar triangles^ =:^, we hate
0a(r) _ 0(r)
an equation for determining the form of the function 0.
Now it is evident that if any function /(r) can be found
which does not change when a (r) is written for r, the
equation
^ =f{r), or ^{r)^r .f(r)
satisfies the above, now Laplace's method explained in
Appendix Art. 398.) affords a general solution of the equation
and thus the complete solution of the problem may be had.
In the particular case proposed, however, the function a (r)
is one of a very singular class of functions, which render
the application of Laplace's method extremely delicate, and
moreover unnecessary. It will be observed that since
r^ + f^*" =x fl"*, therefore
r' = a(r) = ;j/(fl'"- O.
Hence we have o (/) =x « a (r) = a« (r) = ^^"^ - (aT - r^) =
ssr. The function in question is therefore one of those
which may properly be called periodic functions, under which
may be comprehended all which satisfy such, equations as
113
*k*</')=:r, u!^(/')s=:r, &c. and which are possessed of i
Variety of the most elegant and useful properties, which this
is not the place to enumerate. However, it is here to our
purpose to remark, that any symmetrical function of r and
a (r) has the property we wish, viz. that it does not change
by the substitution of a (r) for r, because r becoming a (r),
and a (r), a* (r) or r, these quantities only change places by
this substitution, which, a&^they are similarly involved, does
not alter the value of the function. Let us for instance
suppose (in the proposed Case)
and we fitid
JSimilarly, the equations
tod by ^ a"»r*H-» - f^"» + », &c.
inay be shewn to satisfy the Condition of the problem.
The cases where r + r'=:a, r'" + r'" = a**, r'. r'±=a', w«re
proposed long ago in the Leipsic Acts by John Bernoulli,
at the same time vtrith the celebrated problem, of the Brachy-
stochrone, as a defiance to the mathematical world ; but it
does not appear that their real object, or the point where
their difficulty rested was perceived, either by their proposer,
or by any one of the numerous and eminent geometers whd
gave solutions of them. The attention of mathematicians
being, however, immediately occupied by the extraordinary
controversy of the Bemoullis, and the discoveries of James
relative to Isoperimetrical problems, to which that of the
Brachystochrone gave rise, the present questions were allow-
ed to sink into a degree of oblivion, from which, it will not
be amiss if we attempt to rescue them. They were proposed,
as J, Bernoulli expressly states, with a view of calling the
114
attention of geometers, to a case where the Cartesian methodi
of reducing the conditions of a geometrical problem to art
equation entirely failed, while at the same time the diflferen-
tial calculus afForded no assistance; thus presenting a
difficulty which seemed quite une^cpected, and of a different
kind from any which had yet been felt. This difficulty is in
fact the solution oi 2. functional equation^ or the determination
of an unknown function from an equation, such as those
of (Prob 4.) where it enters under more forms than one, but
Leibnitz, L'Hopital, Newton, and Jas. Bernoulli, all of whom
resolved the problems *, were contented with the first par-
ticular forms of the unknown function which presented
themselves, without attempting to discover any direct process
by which the functional equation might be resolved, and
which in cases of a little greater complexity, constitutes the
only analytical difficulty to be surmounted. -It is rather sur-
prising that this was not observed by Jas. Bernoulli, who
distinctly reduces the problem where r . r'ssa* to the deter-
mination of the form of an equation, which shall remain
unaltered by certain changes made among the variables it
contains. His solution of the problem which requires that
r* . r' shall be invariable, is erroneous, and for a very obvicais
reason, the neglect of the constant in the integral
— , and indeed he himself calls his solution ^' dubiae
X . log X
et suspectae veritatis.'*
The subject was resumed by Clairaut in 1734^ in a
memoir communicated to the Academy of Paris^ in which he
resolves several problems by a method professedly grounded
on, and equival(ent to that employed by Newton in the solu^
* Newton's solution, though extremely elegant, turns on A
peculiarity in the case proposed. It is an application of one of
his own discoveries respecting the sums of the powers of roots of
an equation, and a very happy one^ but the question seems not to
have struck him in the light we are now considering it.
115
tion of Bernoulli's problem above-mentionedj developed,
however, with, great ingenuity, and applied in particular to
one problem of no ordinary difficulty, " To determine the
nature of a curve, such that the intersection of any two of
its tangents which include a given angle, shall always be
found in a given curve." It was in the solution of this
problem, that Clairaut first discovered the class of differential
equations treated of in (Art. 270. of the text) whose general
form is
»-'^=/(^')-
and which has procured him with some, the unmerited praise
of having first discovered the particular solutions of differen-
tial equations ; our countrynian Brook Taylor in 1715 having
deduced the same conclusion in the same manner, and made
the same observation on it *f in integrating the equation
(-'ffX— (^)'
which is evidently a particular case of Clairaut's general
form. Since that period many geometers have occupied
themselves with the solution of problems of the kind in
question, remarkable examples of which may be found in the
writings of Euler, Voss, and Biot.
(5). To determine a class of curves possessed of the
following property : viz. that supposing a system of lines,
ft in number, originating in a fixed point, and terminating
in the curve^ to revotve about this point, making always
equal angles with each other, their sum shall be invariable.
* He diflfereiltiated, and obtained an equation composed of two
factors, one of which leads U^ ^ finalresuU free from differentials,
but containing no arbitrary constant, which is, sjays he " Singularis
quaedam solutio problematis.'* See a clear and impartial statement
of the whole in i;.agrange's Lejons sur le Calcul des fbnctions,
Lect. xvii.
U6
The angle made by one of them (r) with some fixed liao
being 0, those made by the others will be respectively
n n n
Conseqt^ently^ if r = (6) we must have
Q TT Z
n a being some given quantity. Suppose = , then
^ H — ^ = -J!Lli : ^ and so on. If then we n^ko
n n
I -i — I =:«,, we have
The several particular integrals of this equation deprive(^
of its constant term are
2w? ^v z 2(n— 1)tz
cos , cos , . . . . cos — ^
n n n
for the sum of the series
cos -4+cos (-4 + -B)+ cos { -4+ (/I — 1)B }
being
vanishes whenever — ^ is a multiple of ir. Now, if either
of the above cosines be put for u^ in th^ expression
^* T" ^« -+- 1 T" • • • • '^« + H 1 >
a series of this form will arise. These functions then severally
satisfy the equation
M, -}- M, 4. , + . . . . Mj 4. „_ 1 = 0.
117
Of course the complete value of u, in the proposed, or of
4> (B) is
Ug ss a -^ C» . cos 1- C. . cos -f
n n
• • • • ^-^H MB~ B • (fwS
(2/1— 2)?rz.
IS
but rs « , also C,, C,, &c» may be arbitrary functions of
n ' '
cos 2 vz , that is, of cos n d»
Let them be represented by
/, (cos n0)f f, (cos « ff)s &c.
then
0(0) =s r =5 a + cos ^ ,/, (cos « ^ + cos 2 ^ .yi (cos n ^) -f-
....-.+ cos (ii— 1)^ ./^».,(cosii^).
It is easy then to assign an unlimited variety of algebraic
curves which answer the condition. The simplest is that
whose equation is
r = tf 4- 6 • cos Of
which we have already noticed in the case of ;i ss 2, and it
is a very remarkable property of this curve that it answers for
every value of «. In other words : ** In the curve nhose
'* equation is
(x«- *jr + y^y = /j'(a:«+ j/»)
** if a system of any number of radii terminating fn the curvc^
** and making equal angles with each other, be made to revolve
'^ round the origin of the co-ordinates, their sum will be
** invariable throughout the whole extent of the curve."
(6). In the parabola, any straight line being drawn
through the focus meeting fhe curve both ways, the tangents
|t its two extremities include a right angle. To what class
118
of curves does this property, viz* that two tangents so dtzwn
shall include a given angle^ belong.
Let P and P' (Part III. Fig. 4.) be the two points in the
curve, PTQf RQTf the tangents ; MP^MF^ ordinates.
SM=x, SAf^x', SP^r, MSP^e, MP^y, M'P'rry.'
Then, since z.PQP'=(2rr + (2rr=PrS+ PTS,
and that tan PTS = ^, and tan P' T S = - ^', because
ax ax
it lies on the contrary side of the axis of Ae absciss^i diere*
fore if we suppose PQ^P'^ Ay we have
dy _dy^
J dx dx' /_\
tan A =: T 5--/ ^^^'
dx' dx'
Now, the points P, P', lying both in the curve, ~ must
dx'
be the same function of ^ + «• that ~ is of ^, because the
dx ^
same equations belong to both, hence if we suppose
1H =: 0(^), we have ^ = 0(^ + ^)
dx dx
and supposing 2 s:-. and u^^ tp (d) the equation (a) becomes
tan A = — i5 iXi-,
1 4- w,w, + t
or
«*»+ 1 '^« "- cotan -4. . (w,4. , — Ma) + 1 as 0.
This equation we have already integrated (Sect. 4. 20.)
:lhd by applying the formula there obtained, we get
»,=s tan (Jl 2: 4- tan — * C)
that is, C being replaced as before by an arbitrary function
of cos 2 9r 2:,
(0) = tan [ ^ 9 + tan - '/(cos 2 0)1 ; {b).
119
Now we have y^s^r^ sin 6, and x=:r .cos 0, which gives
dy ^dr . cos d + rdd , cos B
dx dr .cos 6 — rdd . sin 6
which put equal to ^ (B) gives a difierential equation between
r and 0, viz.
^ _ cos ^ + » W > sin a ,
r ~" ^ (6) . cos 6— sin ^
Now the equation (A) gives
tan — ^+/(co8 2«)
«W = ^^-J~
1 — tan:2.a./(cos2^)
IT
This value of 4^(fi) substituted in the dif&rential equation
gives after all reductions
log5 ^fd . cot 1^ tan "'/(cos 2 ^) - !!LZL^ ^|
which is the polar equation of the curve sought. Suppose^
to take a particular case, y*(cos Q6)ssOy and- we have
logr=/i?6.cot(:i^^)
and consequently
ssa . < sin $ >
which always gives algebraic curves when the angle A in-
cluded between the tangents is commensurate to the whole
circumference. Thus if u^ = - , we have — ^^^ = — ^ ,
2 «• 2
and
r =:
o™iy
120
the equation of a parabola, the origin of the co-ordin^t€f9
being in the focus.
I{ ji =zw^ or the tangents at opposite extremities of the
line are parallel^ the general equation gives
log 1 =/de ./(^cos 2$),
because the form of the function / being arbitrary we may
replace cot tan ""*y (cos 2 B) by y*(co8 9,6) without infringing
on the generality of the equation. The equation in this form
includes, 1st the logarithmic spiral, by makingyCcos 2 0)=:i^
and 2dly all curves consisting of four similar parts arranged
in the manner of quadfants round a centef. The readet
will find other solutions of this problem by Messrs. Wallace
and Ivory in Leybourne's Repository, (New Series Quest.
l72.) which are well worthy his attention.
Euler, and more lately Ivory, in another solution of this
problem (Thomson's Annals of Philosophy, Oct. 1816.) have
shewn that it admits no solution unless in the cases when the
tangents are parallel or include a right angle, but this limita-
tion arises from the assumed condition that the straight line
PS P^ shall not cut the curve in more than two points. If
we admit, however, that the points P, P\ may lie in different
branches of the same curve, our solution above will apply.
Mr. Ivory's final equation is (in the general case) a functional
equation of the form F (<t> x, <pax) sb 0, in which a? x =r x,
and where the function F is not symmetrical. This he pro-
perly remarks is an impossible equation. If, however, we
admit that different values of the function ^ arising from
radicals, &c. involved in it, may be used in ^o: and in 0a^,
the impossibility vanishes and the equation may be satisfied.
This remark on the nature of such equations abstractly con-
sidered, is due to Mr. Babbage. The problem just solved
affords an illustration of its geometrical signification.
121
(7). ^ In the parabob) the two tangents PQ, P' Q, (See
Part III. Fig. 4.) always intersect in the directrix. To gene-
ralize this property, or to find a class of curves such that
tangents drawn at opposite extremities of any line PSP'
passing through a given point S shall always meet in a
straight line given in position*
Let AQhe the stristight line, and let S A perpendicular to
it be taken as the axis of the x^ then retaining the construc-
tion and denominations of the last problem we have, sup-
posing SA = X and AQ ^Y.
ay dy
Consequently AQn— ~ .AT, or
Oi X
ax
Similarly we should obtain
and equating, these we find
jr =
d^_d_y
dx' d X
Now the condition of the problem requires that this shall
be constant. Denoting it then by 2 a, we get
or,
(2 « - *') . ^ + !/' = (2 a - *) . i? + y-
9
122
Suppose now
(2fl-^).^ + y^u,,
z being some quantity, which changes to z + 1 when x^ yj
change to /, y', then we shall have
Wr+, = w,, or A tt, = O
and of course
u^= constant.
But since the points P, P' are so related that (by similar
triangles)
Therefore the function 2 does not change by the change of
X, y, to x'^i/tor of ztoz-b^t hence the constant in the above
equation may be a function of ^ and denoting this by
X
(2a-^).^ + j/=y'(0; (b).
This equation is integrable at once, by putting ^ = t/ which
X
gives
dx du ^
whence
log — -^ — ^^a,t ^^^-7r-\ = log r, (r)
123
9 />
Suppose, for instance,/ (w)= — — ; then we hare
■•baMi^B».<
(w^ + 1) = r%
(2 a — x)^
or, replacing the value of u ai\^ reducing
j?» + y* = ^ (2 a - *)%
which is the general equation of a contc section about the
focus, and it is easily seen that the straight line in which the
tangents meet, is no other than what is called the directrix
in some treatises on conic sections. The conic section itself
will be an ellipse, parabola or hyperbola, according as the
angle between the tangents is acute, right, or obtuse.
The coiiic sections also satisfy the conditions of the
problem in another way, which, taken in conjunction with
what has just been proved, may be considered as afibrdin|;
a very elegant property of these curves.
If we assume /(») = + -— , We find
%a — a: 2
which reduced, as before, gives
3/* — j' = ^' (2 fl - xY.
This is likewise the general equation of the conk sections,
but whereas in the former case the origin of the co-ordinates
was in the focus, and the straight line in which the tangents
meet, the directrix ; in this it is just the reverse. The origin
of the co-ordinates being now in the intersection of the axis
with the directrix, and the tangents meeting always in a line
drawn through the focus at right angles to the axis (that is,
in the latus rectum indefinitely produced.) The reader may
consult the Mathematical Repository, iii. p. 39- Quest. 267-
for another solution of this question by Mr. Lowry.
124
The final equation (c) of tUs problem presients a peculi"*
arity which ought to be remarked. It is eiddent that by
properly assuming the form of the function y*(M), the integral
in the first member may be made to have any form we
please, and therefore the equation may express any con-
ceivable relation between x and u, or x and y» Yet it ia
equally obvious, that it is not every possible curve which
satisfies the conditions of the problem, but only those of a
certain class. The function ^(m) then cannot be absolutely
arbitrary, but must be subject to certain limitations. Never-
th^lesSy'if we recur to the equation (I) in which the function
y was first introduced, we see no obvious reason for admitting
any limitation of its generality; for the first member is
merely the analytical expression for the distance JIQ (which
is easily proved) and as the second remains unaltered so long
as the ratio of x to y remains the same, or the point f lies
in the same straight line with P and /S, this equation appears
to be nothing more than a mere translation of the condition
of the question into algebraic language. The elucidation of
this delicate point depends upon the theory of eliminations.
Whatever may be the nature of a curve, if we p^t - = k,
we may eliminate either j^ or x between this equation and
that of the curve, and thus both x and y are expressible in
functions of </. For the same reason ~^ is so expressibJo,
ax
and therefore the first member of (b) is in all cases reducible,
by the theory of elimination to a function of « or ? . But
X
it does not thence follow that every curve possesses the pro-
perty in question, for this function may have several values,
and in all the excepted cases actually has so. It appears
therefore that we are not at liberty, in assuming f(u\ to
substitute any form susceptible of more than one value, but
provided this limitation be attended to, it is in all other
respects arbitrary. If we put therefore for^i only rational
125
functions^ we are always sure to arrive at satisfactory
solutions^ but in all other cases it islndispensably necessary
to try^ the solution obtained before it can be relied on. Con-
siderations of this> or a similar kind, apply to most problems
of the nature now under examination, and will obviate any
objections arising from the necessity of limiting functions
which seem at their first introduction perfectly arbitrary.
(8). Required the class of curves which possess the
following property, that any ordinate PM (Part III. fig. 5.)
being erected, and normal MP^ drawn, and at the foot of
this normal, another ordinate PxM^ erected, and another
normal M^P^ drawn, and so on, then the subnormals
PP^y PxP^j P^P^y &c. shall all be equal to each other
in the same series, however they may differ in different
$eries, arising from a different position of the first ordinate.
Let the abscissae AP &c. &c. be represented by x^^ j:.,
x„ &c. the general term Xg being some certain unknown
function of the rank it holds in the series or of jzr, and let the
ordinates be i/o, t^i, y^^ &c. Then the subnormal
hence.
PP, = y,.^; P,P,=:y.,^^andsoon
ax^ dx^
x,-x^^ !/o*7^' '• -'« ■^■^••J^'
H Xq O Xg
and, in general, whatever be z.
^« + 1 = *« + y* • -; — f
axg
or
Now, by the condition of the problem, the series of 8ub«
normals
s, i}h ti ^y^ Sec
aXo rf JT,
126
are all equal, therefore this equation is to be integrated on
the hypothesis of y, . -^ being invariable by the change of z
*****
to jS + 1 . It may therefore be supposed an arbitrary function
of cos 2 ir z or (which comes to the same thing) of tan ^ z^
which let us call Z, then,
A ^, = Z, and A Z = 0,
whence integrating
*, = Z;2 + Z'
Z being another quantity of the same kind^ or another arbi-
trary function of tan it z. But we have
and from these equations, we have only to eliminate z and
we get a differential equation between x^ and j/, (or as we
will now call them, x and y) expressing the nature of the
curve. Now this is easy : for since both Z and 2f are
arbitrary functions of tan nczy we may suppose one an arbi-
trary function of the other.
Let then
Z' =/(Z),
and our equations become
* = Z 2 +/(Z),
The first gives
z = 1^/(2)
Z '
or for Z substituting its value given by the second
=zi-A'-fQ4f)}-
Z
Now Z is an arbitrary function of tan tt 2; ; let then
Z = F(tan TT z)
127
In this equation for Z write its value y -^ , and for z its
value just determined, and we get
^ - ax ^ -^
the differential equation of the curve required, which we
see involves two arbitrary functions. In the particular case
where F denptes an absolute constant r, we have
the equation of a parabola.
(9). At any point of a certain curve, let a normal be drawn
and an ordinate erected, let a second , ordinate be taken equal
to the first subnormal, and let a second normal be drawn.
Required the nature of the curve, that the second subnormal
so determined shall be ^qual to the first ordinate, in other
words, that "in any part of the curve constructing a triangle
whose hypothenuse is the normal, and sides the ordinate and
subnormal, if this be turned into a subcontrary position and
adjusted to fit the curve, its hypothenuse shall still be a
normal.
The subnormal being j/ -j2 , we have the second ordinate
dx
y' equal to y -=-^ . Suppose now we take
CL X
Then will the second sujbnormal, or
= ^ { (y) L= 0' (!/)•
128
v^ \4 .iX^ c^xMtition of the question this is equal to the
- ;c^^ ^^ JuKicir y> hence
.i fcuiH.'tional equation from which the nature of the function
.^ i« to be determined.
Take
y = w., and i>(y)^u,^,
then we have
but the proposed equation gives
0»(j/) = (0M,^, =)y = «,.
Hence we have by subtraction
«, + «-- 0«* =-(!«,+ J- u,)
or,
A i>Ug = — A u,
whence,
01/, 4- w, = constant.
But the two equations
«, =0«, + .» and (pu,=:u,^i
give by cross multiplication
This function, therefore, does not change by the change
of z to ;: + 1, and the constant may therefore bp an arbitrary
function of it, so that
or
from which <p (y) may be found by the ordinary analysis, any
form we please being assigned toy^
129
NoW} integrating the equation,
ax
Ve find
the equation of the curve. Thus if we Suppose f(y.^y)ti
u+iy ,q>(jf) we have
y + (y) - « + ^^ • (y)
if as= js: i^ we have x ss/ydy ss*!. , the equation of i
common parabola, which therefore satisfies the problem, as
does also the cubic parabola.
(10). Required th6 nature of the curv^, such that ah
ordinate being drawn to any point, and also a radius of cur^
yature, a second point may always be found so related to the
firsts that the ordinate at the second point shall be equal to the
radius of curvature at the first, and the radius of curvatute
at the second equal to tlie ordinate at the first.
This problem, by supposing
leads to precisely the same equatioti
from which determining tp (t/), the differential equation above
given suffices to determine the curve.
i$o
«•
If we suppose ^ (j/) ss — which evidently satisfies ^
MI
condition, we have
or, putting -I =;;,
J a^pdp
ydy^ — /_£—
i^lience, restoring the value of p
the equation of an elastic curve. And in the very same
manner might the problem be resolved, if instead of the'
ordinate and radius of curvature, we had ta&en any b^ei^
pair of lines expr683ible in terms of ^, j^^ 7^> &^* ^^
dx a XT
equations to be resolved being, first, the functional equalioil
6f the second order 0* (^) = y, and secondly, a ditferentiat
equation in which the function so determined is involved.
It would be easy to multiply examples of this kind^ but
what we have already given will suffice to indicate th^
method to be pursued in more difficult enquiries of
the same nature. They all lead to functional equations of
greater or less complexity the solution of which is some*
times easily accomplished by reducing them to equations of
differences, though more frequently by considerations pecn^-t
liar to themselves. This problem and similiar ones may be
resolved also by a consideration of the following kind. It is
evident that' since the radiua of curvature at tibe secocul point
is equal to the ordinate at the &8t> aad the vaditu: of xuo^
131
•mature at the first to the ordinate at the second^ thes^two.
functions (the radius of curvature and the ordinate regarded
as functions of x the abscissa) must be such, that when the
ordinate changes to the radius of curvature, the radius of
curvature ^hall qhange to the ordinate, and therefore anjr
symmetrical function of them will remain unchanged. Let
f be the characteristic of such a function, then if JB be the
radius of curvature, it is evident that
f{y^ R) = constant
will satisfy the condition, because this change being made
the equation becomes
/< 12^ ^) =: constant,
which by the peculiar fdrm of/ is identical wit;h the former.
fox R now write its value and we get
•^ y' -dxd*y ) "■
fox the equation of the curve. Thus if we assume
T% ^ / n const.
1/ • it = constant, R = —
which is t^e case just reSolve4-
(11).^ To determine a curve, such that a moveable point
/setting off from agiyenjplaee shall return to the .same place
after being twice reflected against the curve (the aiigle of
incidence being supposed equ'al to tfaatiof rtflection) in iwhdt^
ever direction it first sets offl
hetPAF (Part III. Fig. 6.) be the curve required, S
the given point then if SPP'S be the course of the moving
point and tangents, 8cc. be drawn, we. have, supposing
ISMz=:X, afP=y, :
\. . . ' ^ ■ "
tan SPM^^
. y
tan ikf Pr = -7- , v decr^easmg as x increases,
4y
133
whence we get
tan SPT^tin{MPT^SPM)
= tan VPHf
1 + ^--T^
X ax
or, putting -^ ^pf
O X
tan VPH = y.ZLP±
X +py
by the condition of the first reflexion. But we also have,
tznVPM^:^^,
dy
and therefore
tan HPM = tan {VPH - VPM)
^ xil-^p-)+2py
But, if we draw P'N perpendicular to PM produced,
we have tan HPM^tza RPN :^ »z/^^w. =
— ^^^ , y' being made negative because it lies on the other
•7 """ if
side of the axis. Hence we must have
x(l -/?') + Spy
(a).
If we regard y, ^/^ and x, x , as the successive values of
two functions y and x, of a certain independent variable z,
which changes to % + 1 when the point P changes to P', and
if we suppose the fraction in the second member of this
equation equal to P the equation becomes
133
This equation alone is not sufficient^ as it contains two
unknown functions x and y. To obtain another We must
consider that by what was before proved
PscotanHPJM,
and therefore P' (the value of P corresponding to 2 + 1)
will be cotan HP' M. But the condition of the question
requires P, if, and P* to lie in one straight line, conse-
quently HP M = HTM and
P' ^P, F-PrsAPrrO,
which integrated gives
P s= const = funct (tan irz) = Z; (c).
Substituting this in (t) it is integrable at once and gives
Z' being another invariable function of z. Now our object
being only to obtain a final equation between x and y, we
have only to eliminate the auxiliary variable z, with which
we have no farther concern, between the equations
y =zx.Z + Z'
Z- y(l -/>")- gjpx
jp(l — /?•) + 2j}y
in which we shall succeed by the same artifice as in Prob. 8.
of this section : for Z and Z^being both arbitrary functions
of tan ir Zy we may suppose one an arbitrary function of .the
other^ or Z' ^f{Z) when we have
yzzx.Z -^fiZ)
in which substituting for Z its value
which is the differential equation of the curve.
134
If tbe arbitrary function be assumed equal to zero, the
equation is that of a circle ; and if constant, that of a cdnic
section about the focus. In the former case no integratim
is required.
If we consider attentively the above solution, we $haU see
diat it contains the general principle on whiph jdiat of aU
such problems depends. Let ns therefore take up the fues?
tion generally.
(12). To determine the nature of a curve from any
property whatever connecting two of its points separated by
a finite interval. (See the figure of the last problem.)
Whatever be the nature of the property^ it must enable
us when, one of the points P is assumed, and the figure of
the curve known, to determine the other, P\ or the direc-
tion- of the line PP', hence the conditiofi of the problem
always enables us to express the cotangent of the angle MPBf
by some function more or less complicated of the co-
ordinates at the points P, P', and theif differential coefficients
such as
which we will call P.
Hence by a reasoning precisely similar to that of ijie last
problem we get
y' — J/ = P . (j/ - a?) ; or Ay = P . A J.
But since the property is common to the two points and
connects them with each other, the same^ co-tangent deter-
mined by setting out from the point P* will be represented
by P' and may be had Ixom P by changing a?, y, to a/, y',
and vice versa. But cotan M' P' H =i cotan MP H, so that
P'= P, or AP = 0.
i3&
Consequently P must be regarded as invariable^ and we have
as before
^ = Px+/(P) {a),
»
The problem now divides itself into two caseSi Ist when
the function P involves only the co-ordinates of the point
P and their differentials ; and 2dly, when those of both the
points concerned are combined in it. Id the former case
the equation (a) containing only x^ y^ and their differentials,
IS itself the final differential equation of the curve Sought.
tn the latter, however, another process is requisite. The.
equations
y -y^Pisf ^x)i
y = Pa?+/(P),
P'^P,
inust be combined to eliminate both x' and \f and the result-*
ing equation will express the nature of die curve*
(13). For instance, suppose the relation of the two points
P and P' such that a line drawn perpendicular to the curve
at either of them, shall pass through the other.
Here P HUz normal to the curve, and therefore.
Therefore the differential equation to the curve is
3, = :^ + ftmct(:ii),
or
* +py^/(p)-
136
If we take/(/?) = 0, we have
«• + ^ = r»
the equation of a circle.
(14). Required the ilatiire of the curve in whidi if MH
be always taken equal to ^"^ ^^ xy ; the points P^
P' shall be convertible^ that is^ that M' H shall equal
Since P =^, wehavft P * iir±£|LLL)i
therefore the equation P' = P gives
xy -{■ ay' -{- ab =: Jc^j/ -f^y + ab,
a(2f --y)^ :t'y' ^xy.
Again the equation y' — y = (a/ — «) . P gives
a(j^-j,)=(/-^).iiL±.^«l±±>;....(a>.
whence
aa/y — axy ss xx^y — a:*y 4- a«'y' — aa^y' + a6(«' — «)^
that is^
«(/— y)'^ = (a^y + ab){x* ^x).
This combined with the equation (a) gives
_ (a- x)(xy +a^)
whence it is easy to obtain
137
These values give
ax
which substituted in the equation
y^Px +/(P)
gives
y(a--x)=:ab +a.f(^P)
or simpler changing the form of the function ^
,(,-.) =/(aLti.').
If we suppose the arbitrary function constant and equal
to c% we get y(a - x) =• c% the equation of an hyperbola.
SECTION XL
On Circulating Equations.
(1). To find an analytical function of x, which when
:t is made to pass in succession through all integer values
from to infinity, shall assume in regular periodical rotation
the n values a^ bj c...,kf a, i, c....k, 8ic.
Let a, /?, 7 .... V be the «*•* roots of unity, and let
o a' + /?' +7* +1/'
n
then If we take
Pjp= a • Sg -^ b . Sg^i -j- ^« Sjf^n-hi9
Pj, will be the function required. The reason is obvious,
when x is a multiple of n, the function S, becomes unity by
reason of the property of the roots of unity, demonstrated
in works on Algebra, but in all other cases its value is zero.
*s
138
Now some one of the values x^ j?— 1,. . . .\r~« + l, is ne-'
cessarily such a multiple, and x being made to vary from O
to infinity, this one will be either x, a?— l,....a? — «+ 1,
XfX^l, ....&c. in rotation, so that the function P, will
reduce itself to a, b^, . , .k, &c. in the same succession, and
is therefore the function required.
(2). To find a function P, which shall assume in regular
periodical succession the same values as those of n other
given functions a,, b,, c,, . , .k,. I say that
P,= a, ,Sg+ bg. 5,— 1 + ... .kg. Sg^jt^ !•
For, the values of P, corresponding tox = 0, 1, 2, ... .
7i — 1, «, « + 1, . . . .&c. are respectively a^, ft,, r^, . . . .
i« — 1, a„, bn^if &c.
The functions described in the above paragraphs are
** circulating functions," and may be distinguished into those
with either constant or variable coefficients, of which we have
here instances.
(3). Theorem. Any symmetrical function of *S„ 5,«.„
.. ..<S,_„4.j is invariable. For when x varies from O to
00 , some one of the values of these expressions is always
unity and the rest zero, and, the function in question being
symmetrical, it is no matter in what ' order this takes place,
the order of its elements being of no consequence. The
function therefore has the same value, whichever of its
elements becomes unity, the rest being all zero. That is,
it is invariable by the variation of x from integer to integer
values. Thus
Sg + «J,_ , + aS^t — a+ • • •• '3, -.„_!- J = 1,
(4). Every symmetrical function of the circulating func-
tion^ Pgj P, — „ P,— n+i is in like manner invariable^
provided the coefficients of P„ &c, be constant.
139
For every such function is a symmetrical function of
S^ 'S,_^, Sg^n-hif as will appear if we consider that
by reason of the properties of the roots of unity, we have
S,^n = S,f S'^—w-.j = 5,-5, &c. and consequently,
-Lx — I == k • Sx + tf,0,.^i+.. ..y.Ojp H + l
X^;r— a ^y . O, + i . 0,_i+ ....«• Sx^n+ i
Now any symmetrical function of the second members of
these equations will obviously involve S,, 5, _ „ . . . . 5, _ « + 1>
symmetrically, and will therefore be invariable. Its value
also will evidently be equal to that of a function similarly
composed of the coefficients a^ b^ &c. Thus for instance, if
Px -zza .Sx + ft . Sx^ J, (w being =5 2) we have,
(5). For instances we may take
Px + P* — . 1 + . . . . Px — n + i = a-r^H-^ + .-..^; Sec.
(6). Circulating equations are those whose coefficients
are circulating functions. To resolve them they must be
reduced to others, whose coefficients are of the ordinary
form. The preceding propositions enable us to do this.
To begin with a simple instance, let
Ux + ^ ± Px'Ux+, ± w, = 0,
where P, is a circulating function of the second degree (or in
which « = 2, Px:=:a . Sx ■\' b. S,^^.
140
Assume w, = v, . \/{Px)9
then will «,+, =,v,^., v/(-P* + = v,+,.v^(P,), and the
equation becomes
or
Now the coefficient of the second term \/{Pg Pg + i) being
a symmetrical function, is invariable hj (4) and equal to
>/(tf ^). We have, therefore,
an ordinary equation with constant coefficients, and easily
integrated,
(7)* A more general process however, and applicable to
all circulating equations is to assume for the independent
variable, a circulating function with unknown and variable
coefficients, as in the following equation.
Assume Ujt = A, , S, + Bg . S,..!, and we have by sub-
stitution and by Art. 5. of this Section,
A,. S:, + B,. S,_j ^
A,. S:, + B,. S,_j
whence, equating to zero the coefficients of S, and Sg^^
separately, we obtain
A, +£i.B,«i + a = 0; B, + *. il,-i + ^ = 0.
Eliminating JB,, we find
Ag- ab.A,^^+(a^ al3)z=:0,
141
whence^ A„ is found, equal to
and thence we derive by the second of the above equations,
If these be substituted in the expression for <^^ we get
1 -a* '
which contsuns (as it ought) only one arbitrary constant C.
(8). Suppose the equation were
u, + (a» Sg + b. S,—,) tt,— ., 4- f ss 0.
Here r ss^ . 5, +c. S«-.| ; therefore this is only a particular
case of the preceding, and so of any other constant coefficient
in a circulating equation. This gives, consequently,
— c .
1 - ai
(ff). Let the proposed equation be
11,4. t — R.Uj,=: P,,
where R is constant, and P, any circulating function the
period of circulation being 71, or
Pg = a • o, + i • *S,_i + . . . . ^ . o,««„^. J,
(10). Let the equation to be integrated be
142
where R is constant, and P, any circulating function of the
form
Assume
Ug ^ jig , Ss + Bg , Sg^i + ., . .Kg , Sg^t^ ^ 1,
then
and the equation becomes by substitution
0=(^Bg^^^RAg^a)Sg'^{Cg + i-'R.Bg + b)Sg^,+
-• + (jigJ^i^R'Kg + k) Sg^n-hl
whose terms severally equated to zero give
Bg ^ I ziz R jdg — a^
0,4-1 zz RBg — b^
Ag + i^ R Kg — i,
whence we get
Jg + n=' R"" Ag ^ (a R!"-' +^ /£'*-*+ h).
This last equation integrated gives
A'^R'.{CSg + C . Sg^, + C . 5,««+ , I
1 - iJ» '
C, CV • • • being n arbitrary constants. Now since we have
Ug =: Ag • Sg + Bg » Sg 1 + &c. J
we may neglect in the value of Aj, all the terms but that mul-
tiplied by Sg, in Bg all but that multiplied by 5^—1 &c.
143
which comes ultimately to the same as making C, C", &c.
= 0, because u, can only contain one arbitrary constant C*.
This done we get (putting Q for the constant part of the
value of Ajf),
-Ag ss C> • /w' iSg — Of
JB,= C . R' S,_i - (RQ + a),
K,= C .R' S,^^ + i - {R'-'Q + a R"-* + b R'-\ . . . +J)
which substituted give
. K — ^ _ jgn '^'^
^ B" — ^ + kR •{• a r.
r:rRn ^'-" + ^•
(11). Let the equation be
Pg and P', being respectively equal to
a Sg ^ b , Sg^^i -{■ . . . .^ . Ojp — j» + i>
and
Making the same substitution for u,, the equations for deter-
mining A,f &c. will be '
* The same result will be obtained, if we retain all the con-
stants and investigate in general the values of Ag, Bgf,...,,Kg, If
these be then substituted in the expression for u,^ the super-
numerary constants will all destroy each other> of which we have
already seen an instance in Number 1, of this Section.
144
These give
^ C a ab ab .. ..kj
which integrated gives
A,:=z{abc...,kf. { C. S, + C'-S,^,-i-&c. ^
"" A. ^ A.
u ab ab, . , ,k , t ^ 2.\
_ .{abc k)
1 — ab c, . . ,K
and putting Q for the constant part, and N for (/i • i. . • . ft)"
and^ efiacing all the arbitrary constants but the first, as in the
last number,
jff, = C.N'-' S,.,- (a (2 +«),
C,= C.£i*]Sr'-'S,., -.(a*.Q + fta 4-/3);&c.
which substituted give
t^, = C. {AT'.S, + ^/^x-.5^_^^
l~iVVfl ab ab . , . .k/
\
I
145
1— JVV6 be bc.,,.ka7
!•
(12% It is sometimes necessary to reduce two or more
circulating functions with different periods of circulatibn to
9 common period. This is easily accomplished ; suppose for
instance the functions were
tf . 5, + b . Sg^ii (where «=2),
and
a . S, + /3 . S,_i + 7 . S,— a, (where n=3).
To denote these and similar functions more readily^ sup-
pose we take
^ (n) _ sum of x^ powers of n^^ roots of unity
n
Then will the two functions in question be represented
by
The first of these is obviously equal to
-ha.S,-/^ + ft.5,.
-s
(6)
Because when either x, or a;— ^, or .r— 4, is a multiple of 6,
it is so of 9,y and therefore « is a multiple of 2, conse-
quently, both functions reduce themselves to a ; again, if
ir— 1, or J? - 3, or j:— 5, be a multiple of 6, and therefore of
£, X— 1 must necessarily be a multiple of 2, so that both
reduce themsrives to b. In like manner it may.be proved
that the other given function is identical with
* T
146
80 that the two are thus reduced to the common period
6=?2X3.
tf the separate periods have a common measure^ the
compound period will be their producti divided by this
common measure. The reason is obvious.
By this means should equations occur involving circula-
ting functions with different periods^ they may be integrated.
(13). The following general property of circulating
functions may be mentioned in addition to those enumerated
in 3, 4, 5.
Let
Then
or aty function of a circulating function is itself a circulating
function^ whose coefficients are similar functions of those of the
original^one respective^. In like manner, if P'^^, P"„ &c. be
other circulating functions,
/(P„ P'„...)=/K, ^',,.;.).S,+/(^x, ^„...)5,-i + &c.
Thus for instance (if the coefficients be constant)
we shall have occasion to recal this principle hereafter. It is
too evident to require a formal demonstration.
(14). To determine the integrals 2 S, and 2 P,. '
Putting 2 S, = w„ we have w, + 1 — «*,= -S,, wd this may
be treated as a circulating equation ; thus asdumtng^
147
we have
s (B,^, ^As^ 1)5, + (C,+ . - B,) S,-, + . . . .
(^M-^-i — ^*) 5,— ^.^1,
whose terms severally equated to zero give
JB,4., = 14-^,
C,+, s= B,
whence we get
A particular solution of this will suffice for our purpose, and
it is evident that A^ss ^ v^ill satisfy it. This gives
n
J?, = ^-=^ 4- 1; a = ^-^4- l,....&c.
n n
and finally
w,ss- {x .S,r\- (x + «— ljS,_^ + (a:+ii-2)S,_,
n
(x + 1) S,«^» 4. , } 4- const.
= const + -(5, + S^—i + . . . . 5^^»4. ,)
n -r <
s5 const + ^ 4- - { (»— J)S,-.» 4-(«-2)S,_,
ft ft
.... 4" 1 • ^x»— n + 1 I •
148
Hence, if P,= fl . S,+*.'S,-,+ . ...k. S,-»+.„ we get
X P. = const + (tf + ^ + ^) . -
n
9
+ 1 J («- 1) . a+0 . 6 + . . . . (»-2) • * \ ^*-i
n
+ I {(n— 2). «+(«-!). *+0.^+ J S,— ,
+ i { 1. a + 2. *+.... (w-l)./+0.i J S*-i,+r
»
SECTION XII.
O/* continued Fractions.
(I). To determine the value of the continued fraction^
^ r
«, + — . c
a^ +- , ^.
«#
or, as it may more conveniently be written *,
£i^ f; ^ f^^
«1 + ^a + <»3 + «x
Let.the fraction be put equal to u,. Then we have
Mi = - , Wa = i = ;— ,
* After the example of Burmann-
149
and so on. It appears then, that u, is reducible to the form
2V
— t^N, and Z), being the numerator and denominator of a
certain rational fraction, each composed of combinations of
^\9 ^s» &c. r,| c^j 8cc. formed by multiplication and addition.
Let us now examine them more closely. To this effect we have
c
I
__ Ci gfl . ^3 + g| . C3
., JL ^«^3 (^1 ^* + ^«) ^a + ^1 • ^3
»»1 T —
_ (g, g, . fl, 4- ^1 . 0^4 + ^1 ^9 ' ^4
(«l «« «3 + ^1^3 ^- ^« ^3) «4 + (^1 ^« + ^•)^4
and so on. Hence, we have the following series of equations :
^1 = ^1
JV^S = ^1 • ^3 + ^1 «• • ^3
N^ as f 1 fla . f^ + (f 1 . ^3 + f , flg • ^3) ^4 5 &C.
that is,
^4 sr f^ . N. + /»4.^3
JV, + ^ = c^ + ^.Ns + a, + ,.iV,+ ,; '(1).
Similarly, for the denominators, we have
Di = flj, jD« = f a + oi a^
2)3 .= fli . Tj + (r, + fli tf.) . ^3
I>^ + a = fl, + ajD, + i + r^ + ,i>,; (2).
150
The integration of the equations (1) and (2) will therefore
lead to the values of N, and Dj,, and therefore to that of their
quotient u,. As the8| two equations are precisely the same,
the complete integral of one is also that of the other, and the
values of Ng and Dg of course can only differ by reason of
the different values of the arbitrary constants which enter
into their expressions.
The equations (1) and (2) were noticed at the first origin of
the theory of continued fractions, by Wallis in his Arithmetica
Infinitorum, Prop. 191. p. 192. (Opera Wallisii. Oxon. 1657.)
as rules for the ready computation of the approximating
limits of infinite fractions of this kind, for which purpose
they are well adapted, as they enable us to deduce the suc-
cessive numerators and denominators of the limiting vulgar
N N N
fractions W> 7^' -77 > ^t^- one from the other very readily*
Ui U^ JJ^
(£). . Required the value of the continued fraction
: — (to X terms).
Here c, and a, being constant, the integral of the equation
is N,= C ; o' + C . ^% o and (B being the roots of z^^az + c.
This integral may be expressed more conveniently for the
present purpose by changing the arbitrary constants C and C
c c'
into C + - , and C + ^ which does not diminish the sene-
a p **
rality of the equation*, but only reduces it to the more sym-
metrical form
Ng = C(a' + /3') + c^(a*-» + /3'-»).
* This change of the arbitrary constants might have been
made with advantage in Art. 393. Appendix, where it would
have dispensed with a good deal of pretty abstract reasoning, but
it did not occur a( the time.
151
Determining the constants then so that N^ =i c, N, = a e,
we get
and since Di = a, and D« = a"* + r, we find in like manner
a* + 4c
So that the general expression for f/« is
Let 17 represent the fraction continued to infinity9 and
suppose a the greater of the roots of the equation z^—'a z—c
=0, (without regiard to its sign,) and we get by making x
infinite
which, (as may easily be proved,) is one of the roots of the
quadratic
U* + aU — czzO.
Thus
1 11 , . - \/5 — 1
-— ad wK = ,
1 + 1 + 1 + &c. -^ 2
(to X terms) =
_ 2^ + '4-2("l)*-^^
1+ 1 + 2' + *4-(-l)'
and to infinity =1.
15S
(3). Required the value of the continued fraction
u, = - — - — &c.
continued to x terms^ that is, containing x fractional terms*
The period of the denominators being a^ b, we may
assume r,=s 1, a^=i S, + a 5,_i, where S, is the sum of the
Jt^ powers of the roots of a;'— 1 =sO, and we have by (Art. 1.)
JV, + , - iV, = (* S,+ a S,_0^*+ .•
This equation is the same with that integrated in 6, Sect. 1 1*
and taking
N, rs V, . v/(* . S, + fl . 5,-.i)
= i?,.(v/*.S, +v/^.S,_,), by 12, Sect. 11.
we have seen that v, is given by the equation
Let then «, /5,. be the two roots of
z' — ^/(a ft) . z - 1 = 0,
and we have
iV,= {V^.S,-.i+v/^. S, IKa' + ^+^'Ca'-^ + ZJ'-Ol-
To determine the constants we have ^1=1, N^nb, and
since the value of D, is precisely similar, the constants only
being determined by making D^zza, i), = a i -f I, we obtain
after substitution and all reductions the following value of
the continued fraction
>/(«ft)K + /3') + 2(a'-' +/3—
)9')^-V^(a*)(a'-»+/3'-»)
As before, let a be the greatest root of the equation
2*— \/(ab) . a — 1 =0, without regard to its sign, and let U
153
tepresent the value of 'the fraction ad infinitum. Then by
making x infinity, we get
which is readily shewn to be a root of the equation
a
by substituting for ^/{a h) its value a .
(4). To determine the value of the fraction
1 + ^' ^'-^..
1 -h 1 4-
c.
...-p,
regarded as a function of x.
If we would employ the preceding investigations we must
regard x as constant^ and assume another independent variable
Zj and another function of it p^^ such that
This gives
If then we enquire by the methods above delivered the
value of the expression
I 4- £l_ Pl^ -P«
1 -r 1 + " 1
in its general form, as a function of z^ and then write in the
result for p^ and its derivative functions, the functions
f,— .,+ 1 and others similarly derived from it as a function of
Zi and finally^put j? q? 1, we have the value required. But
the following process is simpler, and less liable to mistake,
* u
154
Assume u, for the value required. Then
u — 1 4- ^'+^
«» + 1 *** = *** + ^* + W
or, taking
an equation of differences of the second order, the form of
which it will be remarked is precisely that which determines
Ng^x and D,— , in the value -j^ — i of the function
Co, c,^
1 + 1 + 1
(See Prob. 1.)
But though the equation of tiifferences is the same, the
nature of the functions derived from it will be essentially
modified by the different constants required to adapt ^ts
integral to the two cases. Still, however, this coincidence
assigns a relation between the two functions sufBciently
remarkable.
In fact, let {A, -i- C . B,) xC he the general value of v,.
Then will
and
where the constants depend on the values of r^, r,, Ai^ A^
-Bi, jBa, and are easily determined. Now these expressions
are respectively the values of
155
J + 1 + ••••••!
and
1 + 1 + 1
(5). Having given the value of
1 + fl— f^^ni 1^ = F^
1 + 1 + 1
To determine 'that of the same fraction with any additional
denominator at the end^
1 +
Cx ^jr — 1 ^1
1 + 1 + a
If we proceed as above, by putting -il±-^ for it^ value we
Vjf
shall obtain the very same equatioi^ of differences and there-
fore the expressions of . the two functions can only differ in
the values of the arbitrary constant C in the expressicm
"^.^ ' p'*^^ ' Now since one value F^^ of -i±i Is
A, + C . B, V,
given, one value of v^ is also k^own, being equal to
i^i . F,. . . . F,_,. Hence a particular integral of the equa-
tion for Vg is given, and of course, being only of the second
order, the complete integral may be ascertained by the
methods delivered in the text (Append. Art. 382.) : and the
constant must then be "Adapted to the case in question.
(6). To find^ for instance, the value m, of the fraction,
. ,c c c
1 + — " » — ,
1 + 1 -h a
when the letter c occurs x times
156
/9(l+£)-./3.
a and /3 being the roots of v' — « — c =s 0.
SECTION XIII.
Application of the Calculus of Differences to various
Problems.
(1). What are the respective amounts of a given sum
for X years, at simple, and at compound interest ?
Let P, be the amount at the end of the x*** year, then
Isty at simple interest, if A be the original sum, and r tbe
interest of £.1 for 1 year,>^ will.be that of £A, and there-
^fore the increase in one year being rA, P^ + rA is the
amount at the end of the (x+iy" year, but this amount is
alsolrepresented by P, + , . Hence
Ps + i=Ps^ rA, or AP,= r A,
and integrating
P,= r A ,x + C.
Now the original sum or value of P„ when xmO^ is Ab
hence
P. = r ^ . + C = ^ or C = il,
and therefore,
P, = .4 (1 + r jr).
2d, At compound interest. P^ being the capital at the
157
end of the x^ year, r P, is the interest in the {x + 1)***
therefore,
P,+ . = P, +rP, = (1 +.r).P.,
and integrating,
P. = C(l + ry.
Now P, = ^, hence C = 4, and P, = ^ (1 + r)*.
(2). A person places money in the funds^ but gradually
contracting expensive habits, he spends the first year the
whole interest, the second twice that of the remaining stock,
the third three times that of what is left, and so on. How
long Will his property last, and in what year is his expen-
diture greatest ?
As before, let P,=his stock at the end of the x'** year,
r= interest of <£l, for 1 year.
Then r P, = that upon P,, and consequently his expen-
diture in the (j: + 1 )' ^ year is (j? + 1 ) r P,. Therefore at tKe
end of the (x + iy*» year his stock will be
P, + rP,^{x + l)rP, = P,(l -Pfr),
Hence,
P,+ ,=(l-xr)P.,
and integrating on the hypothesis P. = A,
P, = -^1(1 -f)(l-2r) ^{ l-Cx- l).r } .
This vanishes when x = i + - , which is the number of
r
years his stock will last. Also, his expenditure in the x^^
year being xr P,^^^ and in the next, (x + l)r P, will be
greatest just before A (j: r -?,_ ,) becomes negative, because
then having reached its maximum it begins to decrease.
158
Now
A(Xf P,_,)=: 1.(1 -r)
Suppose then
(a? + 1) (1 - r r) - 1' = O,
this gives
"\/(^D-^
and the nearest integer less than this is the required number.
1
If r5= — X x = 4f exactly. Here then A («P,«.g) =: O,
when X =s if, ^ that the sums spent in the 4th and 5th years
are equals and greater than in any other.
(3). A person puts out to interest a sum of money (A),
he expends annually a portion (a) of the interest, and adds
the remainder to the stock* What is the amount after x
years ?
Call it F, ; then the interest is r P,, so that
P*+ 1=^ P, •^-rP,'- a, or
Ps+t - (1 +r)P, + a =0.
which integrated gives
P, =:C(l + r)' + f.
r
The constant C, must be determined by the consideration
that Po the original stock is equal to ji w^^^^ gives
r
and
159
I
If a exceed the interest, and on that supposition we would
find how long the money will last, make P, = 0| and we
get
*
^ _ log a - log(fl ~ rA) ^
log (1 + r)
If we would find what annual sum a stock-holder may
expend so that his property shall just last out his lifey^t>n a
fair calculation, call x the number of years he has a reason-
able expectation of living (calculated from the tables of mor-
tality) then we have
(1 + ry
a ss r A ,
(1 -h ry - 1
(4). A has <£.1000 (= ^) in the funds at 5 per cent,
(ss r). He spends the first year the full interest of his'
capital c£.500, the next <£.1000, and so on in regular arith-
metical progression. How long will his property last ?
As before, putting P, for his property at the end of the
x^^ year, we have
P, + , = P,(1 +r)-(j?+ \).rA,
which integrated gives
Hence we have to find x from the exponential equation,
(1 +r)'= 1 + r + rj,
and since r = — . , x = 6 .G nearly.
Were a more exact value required, we must proceed thus :
Suppose f{x) =3 0, and a being an approximate value of x,
let 41 4- A be the true value, then will h be small, and the
equation
/(^+A)=:0,
160
developed by Taylor's theorem gives, neglecting h*, 8cc.
aa
whence,
, f(a) .da * , . ,
n = -^ ^ , and jr s a + *,
d/(a)
it a second approximation.
In the present case,
ana)
da
= (1 +r)'.log(l + r) -r,
and
(1 +r)Mog(l +r)-r'
= - 0.00453,
so that a second approximation is r = 6.59547.
(5). A man spends every year twice the sum he gained
by a certain business the year before. That business^ how-
ever, becomes every year more and more profitable, and he
finds his property increase regularly^ as the square of the
time since he began business. In what progression do the
profits of his trade increase ?
If we call P, the profit in the x^ year^ the problem leads
to the equation
Px + i- 2P, = ^(2r + 1),
whence we find
P,= a\5.2'-' - (2z + 3)} + P, .«'-'.
Now P.= fl . 1% hence P, = a (3 . 2' — 2 x — 3).
(6). An individual sets out in the world with a certain
capital (il), one-half of which he places in the funds at r
p^r cent, and the rest, venture;s in a concern which produces
161
2 r per cent, per annum^ but which returns the interest only
qpce in two yeaij^ ; he lives at a stated rate of expenditure
{a £ per annum) and puts all his gains and savings into the
funds. Required his funded property after any number (x)
of years.
Call his funded property at the end of the (x^^) year P,.
Then^ if ^ be an even number, the interest on that part of
his stock I — J which is vested in trade does not accrue in
the (j? -f 1)* year, so that, {x being even),
P,+ ,=(l +r)P,-^,
but when x is odd the interest for two years at 2 r per
A A
annum, accrues on the capital — , that is, 4 r - or 2r A,
hence, (x being odd),
P,4.i = (l -^r)Ps- a +2rA.
It may not, perhaps, immediately appear how these equa-
tions are to be treated ; because in either of them, if x be
increased by unity, the equation ceases to be true, and there-
fore the function P, cannot be found by integrating either of
them separately, the law of continuity being broken. To
supply this, and to include the odd and even values of x in
one analysis, we must have recourse to the theory of cir-
culating functions above delivered. In fact, since the cir-
culating function of the second degree
a .S, + (a — 2rA).S,^iy
m
is equal either to a, or to a — 2 r ji, according as x is even
or odd, the equation
(where i?= 1 4-r) includes both the others, and admits of any
value being assigned to x. The law of continuity being
162
thus restored we may find P, by integration, and we get
(9, Sect. 11*)
p.^c.R'^:
aR + a — Q,r A
1 - R'
(a'^*2rji)R + a r.
— 1 ■^:i *^* — :
1 - R' '"^''
and determining C so that Po = | -i, his funded property at
the outset, and writing for S, and S,_, their values
^ "^ ^"" ^^ and Lri-^lll^' / we get finally,
(7). The same being supposed as in the last problem,
only that the part of his capital vested in trade, yields r' per
cent, per annum, but returns interest only once in n years*
Required the amount of his funded stock after any number
of years.
In this case the equation
holds good for every value of x unless when x+l is a mul-
tiple of n (and therefore x — « + 1 such a multiple) when it
changes to
' +
, = i? • Px — (fl *-* h)y
A
where h ^ nr\ — . In this case then we have for our cir-
culating equation,
^ P X _j. , — R , P X + \ a • Ox "Y o, m Sg , 4"
,,,,.. {a — b) Sx^n + i } J
which integrated as in (9, Sect* 11.) gives
163
R - i U" - 1
the constant C being determined as before, hj taking x ^ 0,
is found as follows :
2 iJ--l iJ-l
- T (^ "*■ :r^) " :r^i •
4
(8). Suppose a merchant engaged in more than one
such concern as those described in the two last problems.
To determine his funded property after any time.
L^t n represent the least time in which the interest of
funded capital can be made readily to accrue, n\ n", &c. the
intervals at which the several parts of his capital embarked
in commerce return their interest, the least common measure
of all their intervals n, «', ft\ &c. or of such as differ from
each other, being taken as the unit of time. Also let
jf, A'\ &c. be the several parts of his capital so embarked,
andy, r", 8cc. the rates of interest they yield in the time 1.
Lastly, let A be his original funded capital, P, its amount
after x such units of time have elapsed^ r the rate of interest
in the funds for the time 1, and a his uniform rate of expen-
diture in that time. Then we have
P:r+i = P* — <?>
sinless when x + 1 is a multiple of either /i, ff^ &c. in which
several cases, the terms
nr^P,, n' r' A' &c.
expressing the respective sums accruing as interest at these
moments are to be added to the second member. Employing
164
then the notation of (11, Sect. 11.) the circulating equation
embracing all these cases isj
~a + n' r' A' . 5._,'+ 1<"'' + 8tc.
Take m = the product of n, n', &c. divided by all the
greatest common measures of any two or more of them, and
this equation is transformed to
-P,{ 1 +«r(S,_,+x«"> + «,_.»+ /"'. . . . +«,_„ + .<"')} +«-
- ft" r" A", &c.
Now, since
and
tf = « . S/'") + . S,^^^^ + &c.
If we take
0^=1^ «,= 1,. .. tf»=: 1 + /ir, ^M + i = 1, ^» + a = 1) • • • •
/ign = 1 — /ir, . . . .oCC*
ii s: Oy ^2 = J ^», "^ ti r A J On, ^. I ss Oy
^„. :;=n'r'A', &c.
^1 = 0, ^n" = «" r"^', r»" + 1=0, &c. &c.
and finally
— a + ti + c^+ .. .=a, — a -V ^, 4- Tg + . . . = ^, &c. &c.
our equation will become
- { a. s;-)/?.s,.,c-)....K. «,_,+,(-)} =0,
whose integral is given at full length in (10, Sect. 11.)
165
In this case we have
N = (1 + nr)%
and taking P^ = A, we find for the arbitrary constant,
1 — iV Vai a^a, flj . . . . fl„^
but till the particular values of n, n\ &c. are assigned in
numbers, no farther reductions in the form of this integral
are practicable*
(9). A and B engage in play, on the following condi^*
tion, viz. that whehever A wins a game, the stake shall be
doubled for the next game, but whenever B wins, it shall be
tripled. When they left ofi^ (after x games) it was found
that they had won and lost alternately, A winning the first
game. What are their respective gains and losses ?
Put u, to represent jfs total gain at the end of the x^^
game, and suppose P^to equal the stake for which that game
18 played. Then, provided A win the x^^ game, we have
P,+ . = 2 P„ but if B win it P/+ , = 3 P,. Now A wins
it if X be odd, but B if even ; hence in all cases, the nume-
rical value of X being undetermined,
P,+ . = (3 . S, + 2 . S,^,) P,; (a.)
where 5, =s | the sum of the i*^ powers of the roots of
x;' - 1 = 0.
Again, A u, is A's gain or loss by the event of the
(a? 4- ly** game.. It is evidently equal in value to the stake
for which that game was played \P, + ^) being z gain if A
win, or .r + I be odd> but a loss if he lose, or x + 1 be even,
hence
166
It only remains therefore to integrate the equations (a),
{b). Now the equation (a) is that of (8, Sect. 12.) .whence,
determining the constant by taking P, the first stake for a
given quantity, we find
P, = Pi V2 . f Vl . S. + s/s'.s,^^ ] {\/6y-\
This given, we get, by substituting the value of P,+ i
in(*),
K
To integrate this, we take Us = A,. S^ + B,. S,_i, which
gives by substitution and equation of like terms
^,+ ,-B, =-2P,-(>/6)'— , or
Adding this and the first together, we find
J. + .-X = -P,.(v'6)',
and integrating
consequently
5
and
%
But M, =: 0, hence C s: ^ P, and finally
167
for th^ total amount of ^'s gain or JB's loss at the end of
their play.
(10). The last problem may be generalized by supposing
that when A wins, the stake of the succeeding game shall
become any function whatever of the former stake, and of
the number of games elapsed since the beginning, and when
jB wins, any other functions. A and B winning alternately,
what is the total amount of A^s gain or loss ?
Here we have P, + i =/(?,, x),
when X is odd, and =f^(P^f x) when even, so that in
general
P, + . = S, ./ cP„ x) + S.^, ./(P,, x).
To integrate this, or at least to clear it of its circulating
form, we take
P, :^ Ag . Sg + Jjg • p, ^ J,
Then since jr(P„ x) is a function of S,, iS^ — i it is redu-
cible to the general fornt- of all circulating functions (by
12, Sect. 12.) and in fact it becomes
/(P„ X) =f{Ag, x).Sg +f(B^, X) S_„
similarly,
/i (P*> ^) =/i {A, ^) . Sg +/ (Bg, x).Sg^ 1,
therefore
Ss\Bg^^^A{Ag, x)\ +S,«,^, + i-/(B„ .r)l =0,
which resolves itself into the two
That is,
Ag + ^z:if {f,{Jg, x), X]
168
an equation of the second order whose integration suffices to
give A,. If we put j? = 2z and A^^Ajy this reduces itself
to the first order, after which it depends on the particular
forms of ^ /,, whether the integration be practicable or not.
A very extensive case of integrability is when
In this case the equation for finding A^ is linear of the first
order, viz.
= a^a^ . AJ + (B„ + asM^oM'
In like manner, we may proceed when instead of winning
and losing alternately, the players win and lose in any other
regular order.
(11). Let there be a series of quantities A, P, C, &c.
derived from one another by the following law,
A=:a, ^ = — - — , C = g — , JD = ^ — , &c.
Required the general term of the series.
Call the j:*^ term u,, then the (r + 1)* being w^ + i we
have
a? + 1
(^ + l).ii,^., 4- «a?W;, = 1
an equation of differences which integrated give,
xu,- i + C.(- 1)'
169
and determining C so that u^ = a
( 1 2). Suppose we have
A=:a, B^^A*" 1, C = 2 5* - 1, &c.
Kequired the a*** term of this series.
Here w* + 1 = 2 tt/ — 1,
an equation which integrated gives
the constant being properly determined. In the same way
we might proceed, if we had
A^a, B ^f{A\ C =/(B), D =/(C) 8ic.;
(13). To integrate the difFerential expression '.
/ x*''dx
Assume it equal to F„ (being a function of n to be deter-
mined). Then, integrating by parts.
i;=/j— •^ """^'^
.a'
z= - x^"-' n/(1 —.»*)+ (2 « - 1)/j:**-* dx. s/{l - x«),
1 — a*
or, putting P„ = x**— * v'Cl — x'), and writing -— ,-^,
for n/(1 — •J^*)
That is,
170
an equation of differences {n being the independent variable)
whose integral is
F — ^ ' ^ - ^ (2 HI — 1)
* "^ 2.4 (2w)
1^^^1.3.5 , (211+1) ^"-^^r
The integration denoted by the sign 2 not being practi-
cable we must (as in 3, Sect. 4.) write at full length the
series of which the integral consists, viz.
1 p . ? p . 2 . 4 . . . . (2 « - g) p
1^^ "^ TTs ' ^ ; • * • 1.3....(2«-ir "'
and determining C by the condition
= / -' = sm~*j:.
^» ' v/(l-^*)
we get, restoring the values of Pj, P^, &c.
1.3....(g«-l)i _/x 2^
2.4 (2«) I Vl 1.3
^♦4... (2«-2)^^_A >
1.3 (2«-l) / ^^ ^i
If we only require the value of the integral between the
limits X =: 0, and x =i I, since x"*—* , ^(i — . x«) vanishes at
both the limits, we have P^ =? 0, and our equation of differ-
ences {a) is simply
Pn- — Fn^, =0,
2«
\vhich gives
^ ^ ^ 1.3 (211—1)
2.4 (2 «)
171
but C zi I z= - , between the same limits ; so
that
2.4....(2w) \2/ '
and in the same way may all similar instances in which
integrals are reduced by successive steps to a less and less
degree of complication till at last they are brought to a
known form, be treated without going through the process
of continuation, by reducing them to equations of differences^
EXAMPLES
OF T«E
5olutton0
f>r
FUNCTIONAL EQUATIONS.
H\
CHARLES BABBAGE, a.m. f.ii.s. l.&e. f.c.p.s.
AND SECRETARY TO THE ASTRONOMICAL SOCIETY OF LOra>ON.
.*
NOTICE.
The object of the following Examples of Func-
tional Equations^ is to render a subject of considerable
interest, more accessible to mathematical students^
than it has hitherto been. It is, perhaps^ that subject
of all others^ which most requires the assistance of
particular instances, in order fully to comprehend
the meaning of its symbols, which are of the most
extreme generality ; that assistance is also more
particularly required in this branch of science, in
consequence of its never yet having found its way
into an Elementary Treatise.
Oct. 20. 1820.
OF
FUNCTIONAL EQUATIONS.
If a function a is of such a form, -that, when it is twice
performed on a quantity, the result is the quantity itself, or
if a^ (jv) as X, then it is called a periodic function of the
second order, if a" (x) = a?, then it is termed a periodic func-
tion of the «* order, thus when a (x) = a — .r the second
function, or
If «(x)«-i-i
1 1 — .r X — I
then a*X=:a(ax) =
J _ 1_ 1 — Af- 1
9
1 - a:
and
I
a J7 — 1 1 — j:
- 1
a' J? as a^ a j: sfi — — - — as =1-1— j:=:;v.
1 —X
the first of these examples is a periodic function of the
second, the last is a periodic function of the third order.
Fbob. 1. To find periodic functions of the second order.
Since such functions must satisfy the equation >//• x = x,
we have
or yj/^ must be such a function, that it shall be the same as
its inverse ; if therefore j/=\/^x, we have also <r=>/< — *j/5=^j^,
tA
(2)
or if X and y are connected by some equation, it must be
symmetrical relative to x and y\ y or ^x must then be
determined from the equation
for instance, if a? + yl^x -^ a =^0^ y]/ x ts a — x,
or i{ X yfy X zs a^, y\r x sz — .
X
Another method of determining such functions is as
follows : since \^ a? is of such a form that yl^*x =: x any sym-
metrical function of x and "^x remain constant when* is
changed into >//- x thus
F\ X, yjj-x \ becomes jF{ >/rx, yf/'x \ = jF{ yf^x, x | ,
if therefore, we can find any particular solution of the equa-
tion yj^^x =i X, containing an arbitrary constant we may sub-
stitute such a function for it, but yj/^xssa^x is a particular
solution therefore
yj^x = F{x^ yf/^x) — X,
X + yl^x st F(x^ yly-x),
and by changing ihe arbitrary function into another of the
same form, we find
F^{lcy yl^x J = 0,
as before.
These two methods of determining periodic functions of
the second Order, are not so convenient as a third process
which can be extended to all orders.
* Bars placed above quantities tinder the functional sign, in-
dicate that the function is symmetrical relative to those quantities*
(S)
Assume >/'•« = <^~^f<^ *, then
this must be equal to x or
this eqi]iation will bd fulfilled if /'t>=v, or if /is a particular
solution, and if also 0~~* is such an inverse function that
— * V s= V, If therefore is arbitrary, and /is a particular
solution oi f^x ss x, then the solution of y* a- = a: is
Ex. Let/x=^, then ^^ j: = 0""* (— ) >
J? ' \(p x/
•^^ b^cx' ^ ^ \b-{-c<px/
from these may easily be derived the following periodic
functions of the second orddr,
yf/x z= a — X yj^x
a?- 1
a: — 1 r
>^j?a5— - — x/rj: =x \/l - a;'
1 + ^
. ^ + 1 . a:
X— 1
\/j?*— 1
x^j: = tan^* I ^i L 1 x/^j,- = log (a — i')
Vcos a. cos x/ ^
x^ a: = (a" - *~)'* \/^ x = j: - log (•' - 1)
JT
xZ/'Xs yjxx s= tan "■ * (<j — tan r)
or if X and y are connected by some equation, it must be
symmetrical relative to x and y\ y ox ^x must then be
determined from the equation
*JP{x, ^{ =0,
for instance^ ifa^-fx/^-x — a=0, yj/ x ^ a — ^,
or if X yfy X zs a^, >|r a: =: — .
X
Another method of determining such functions Is as
follows : since >^ J? is of such a form that y}^*x =: x any sym-
metrical function of x and >^ar remain constant when* is
changed into yj/^ x thus
F\ X, yjj-x \ becomes jF { >^x, x/r* j? } = jF { >^ a:, x- | ,
if therefore, we can find any particular solution of the equa-
tion yj^^x = X, containing an arbitrary constant we may sub-
stitute such a function for it, but yf/^xsza^x is a particular
solution therefore
«
\j^x = F(Xf yj^x) — X,
6r
X + yjy-x ^ F{x, ylrx)j
and by changing ihe arbitrary function into another of the
same form, we find
F*{lcy yf/^x J = 0,
as before.
These two methods of determinitig periodic functions of
the second order, are not so convenient as a third process
which can be extended to all orders.
* Bars placed above quantities under the functional sign, in-
dicate that the function is svmmetricai relative to those quantities.
Assume \l/x = <!>—' f^x, then
f
this must be equal to x or
this eqi^ation will bd fulfilled if f^v=:v, or if /is a particular
solution, and if also 0~~' is such an inverse function that
"■ * V = V. If therefore is arbitrary, and /is a particular
solution o{ /* X ss X, then the solution of \[r^ x = x is
Ex. Let/j:=^, then >//■ ar = — ' ( — ) ,
a: " \<p x/
^ + t: J? \0 + c <px/
from these may easily be derived the following periodic
functions of the second ord^j
y\/ X =z a — X yj^x
X - 1
\f^ jr = ^ \/^ J = —
a: — 1 X
1 —a?
"^ X s^
yf/x =
1 + T
X + 1
a:— 1
%//■ j:
=
v/i-
x-"
>/^.^
..^
X
s/j?*— 1
>|. x = tan-> (sin(g>-a)\ _ __
Vcos fli. cos x/
>|^ j: = (a** - Ac")~ ^ \|r j; =r j7 - log (•' — 1)
X
xj^xsz ylrxs=: tan ^ * (<» — tan cT)
(4)
Prob. 2. Required pbriodic functiotis of die tMrd
i>rder, or such as fulfil the equation >/f' « s= dr«
Assume >/ra!s=0—^/0x9 then the equation becomes
which will be verified if f(v) is a particular solution of
y* v«v> and if ^""' is such an inverse value that ^— '^ t;=vt
hence the solution of the equation is
one solution is and hence yUJif ts 0'—* ( .. . 1
more particular cases are
I ^* ,1+4?
a — 0? 1 — S j:
a*
ylfX^-—-r — -— ylrxxs
s/a j^ — a'
ac — e^ X X
, ax '^ a* . 1
yfrXss ^ y^r X sz
X ' \^ X
i
X %» 9
PROB. 3. To find periodic functions of the ri^ order^
or to solve the equation ^""x ^x.
Assume as before ^ j: = 0-»/0a: then it becomes
(5)
which 18 yerified if / is a particular solution of/* x ^ x, and
if "^ ' is such an inverse function that (p—^ <l>x ss x.
It now remains to find particular solutions of yj/'x = x
which may be accomplished in the following manner z let
fx represent
f ormj or
a+bx
c-^dx
then the n^ function will be of the same
/n / \ -^n T -^n X
where A^^ Bn, Cn, Dn% are functions of a^ h c^ d, and n^
these may be bo determined that i)«sO^ il»==0 and B%^Cn
all which conditions are satisfied, if
*V-2ftrco8l*I + c'
d = -
n
(
2 + 2 cos
hence
ipX
=0-0
a + bipx
^»--2Arcosiil + r*
n
c —
(2 + 2 cos ^),
j;i
a more detailed account of this method of solution may be
found in a paper by Mr, Homer in the Annals of Philosophy,
Nov. 1817.
Instances of \/^* x = j? are
I 1 1
y}/ X =:
y}rX = 2
2 1 —^
2
2 — a?
J7— 1
X
yf/^x
=
1_
1
+ x
— a:
y}^ X
3fl«
2
ac -^ c*
X
yj/ X
=:
n + b
j:
c
*• +
c"
X
9,a
(6)
yj/X
v/2 -a?*
(2 j:* - 2)^
>//■ X s= log 2 — iV + log (•* — 1).
All those cases which satisfy the equation \//^'jc=jr, also
fulfil that of yj^^x =: x, as well as all those which fulfil any ^
«| these equations >P j 5= — ar, V'*^ = -» j o>^ «»ore generally
yff^ ^=a J) where ax is a particular solution of the equation
The following particular cases satisfy the equation
1 , Sx— 1
\l^X=: VX=
^ 3 (1 - X) d X
3 , 3«2
^x ^ y X =
3 — X 3/?^ — r'x
^x — 1 , 3 + 3x
yj^x = 3 yx =
^x =
X 3 — X
a + bx
c- —X
3/1
>//X = - I x" I
x\ 3/
\^ X = log 3 — X + log (•' — 1).
The principle on which the solution of the functional
, equation F\x, ^l^ Xy y^/ax \ =0 depends, where a' x=:x, is
that by substituting ax for x we have another equation
F \aXy yf/ ax, xj^ x } = 0, between which and the given
equation we may eliminate yj/ a x and the result will be thq
value of \/rX a few examples will illustrate this method.
(1)
(1)^ Given >K:r) + a >/^ (— a?) = j:*
by putting - ar for a: this becomes '
and eliminating >K— a?), we have
hence
1 — fl»
(2). Given >/rar — a>/r« =1*
1 r 1 '
put - for a:, >//•- — fl( >/r a: = f
a? a:
and
y^rx ^at' — a* >/r a? = 1',
>£/'a: =
1 -a»
(3). Given (>/^a:)\x/.i f = c«^
1 + J?
1 — ^
put for x, it becomes
1 4- X
/ , 1 — ar\' , ^ 1 — ^
eliminating >//• by means of the former, we find
1 -h a?
^^=(t^''*0
JL
3
(8)
(4), Given yf^x + — l_^^(i - x*) = 1 + x'
putting >/(i — x*) for x, we have
J?
and substituting this value o£ >//• y/1 — x* in the formet equa-
tion
Q — :r* 1
1— J?* j:' — X*
hence
and >/r a: = ar*.
(5). Given ^^ +^ ^<-^> =: i
I -{-ylrX 1 +>/,(— x)
put \^, a: = , thus the equation becomes
>/r, X + a: >//•, (— x) = 1, and changing * into — a: we have
^i(— ^) — ^>^i W = l> by which elinunating >/r(— a?) from
the former^ we find
. 1 — X
^' I + X*'
hence
yjr X ss
— ^^^ ^ ^"^
1 ~ ^i «» a: + J?*
(6). Given yj^x-^Ll^ ^- = c.
X X
putting -- for x this becomes
X
(»)
X
aad by eliminating >/^ i, we have
X
^ ^ = ' i c
1 + J" + T*
(7). Given >^ j? -f * \/r (1 ~ i?) =,s 1,
putting 1 —X for x, we have
>^(1 ^X) + (1 -X)ylr(x) = 1,
whence, by elimination,
I - X 1 - X
y^r X =i
1 - j:(i — «) 1 -jr ^- J?^
(8). Giyen -±I- + r—lll-Zf) = i,
yfrX'-X >^(1 — J?) -I- X — 1
put \/r- x as -T-^f- , then Mrill >//•,(! — x)— . — ,
and the equation becomes
\l^^X + X\fr^{l - X) =s 1,
the same as in the last example; let /x represent the solution
there found, then
whence
/x- 1
1 — X
if we take for fx its value -, we have
•f B
(10)
In case the equation is symmetrical with regard to \^ or
and yj/aXj the process of elimination apparently becomes
illusory. By a peculiar artifice this difficulty may be over-
come, and it happens rather §ihj^Ulatly that iii 01 thiiSe t^k^^
the solution which is so obtained contains an arbitrary func-
tion, and in general the solution is the most extensive which
the question admits of.
(9). Given \//^ x = x//' - .
If we put - f<5r r, thii is changed into \/^ - = x//- x, the same
as the given equation ; it is therefore impossible to eliminate.
Let us now supposb yl^k'=:Q^^-{'h,
X
which becomes the given equation when ^ e 1 and ( 3=; 0.
By putting ^ for t this is changed into
x//- - sz a \l^x -f />,
.r
and eliminating \J^ -, we have
, ^ _ab -hi b
V^ = = ,
I — a* 1 — a
if ^=0 and £r = l, this becomes a vanishing fraction whose
value is any constant quantity r, and we have yf^xsssCf which
fulfils the equation. This is a very limited solution^ but the
following plan will lead us to much more general ones.
Take the equation •
4rj: ^ a yj/^ ^ -\- V d> x^
X
which coincides with the given one when v=Oanddr= 1 ; also
^ X is any arbitrary function of x ; putting - for x, we have
X
/
(U)
and by elimination,
S^ X
1
tf0- 4- ^a^
I - «'
V.
L6t a become 1 + and v become at the same time,
then
O 1 J
l—l.l +2.0 + 0') -2. 0+0* -2+0 2
and tlie solution becomes
1
or changing the arbitrary function
\L X = (px -^ fp - ,
X
in which <p is indefinite.
This solution is, in fact, nothing more than an arbitrary
symmetrical function of x and ~ , and may be expressed thus
^x = xQ^l).
Precisely the same course of reasoning will produce the
solutions of the following equation.
(10). >/. (J?) = >/^ (a - x)
xj^X z=: xi^Vy £J — X).
(10. V. (.) = ^^ (Lz5) , V'.=x{l,f^}.
(.12)
(.2). v^ (,)=,/, (_^^),>^, =4^' ;Ai^)I
(13)
• ^(^)='^^^-'>' ^* = ^{^'ttI|
(14). >//- x = >/^ (o x), >^ A' = X (^* <* ^)* where a* x = jr.
(15). The objection which has just beexvstated occurs
in the equation yf^j-^yf^ ( j = c,
and a similar mode of proceeding will obviate it. The given
equation is a particular case of
with which it coincides^ if a = 1 and vssO ; putting for
X— 1
X in this, we have
and elimination produces
«•
If v=:0 and €i = l, this gives
in which the function ^ has been changed into another simi-
lar one.
16. Given \/^(l + ^) + x/^ (1 — ^) = 1 - x*,
put x— 1 for 1, then
f
(13)
x^ JT + VK^ — a:) = 1 - (x - !)• = 2 X - ^
this is a particular case of the equation
with which it agrees if v=:0 and ar:! ; changing x into 2— jr
and eliminating >//* (2 — x) from the result^ we find
^^ = z ^^ + { 0*-^ 0(2-«) { .-
f 1 ~ a' 1 — a*
or
If a=l and v=:0, we have
2 ,r — - ^*
Vx J? = --, — -f- X - (2 - a:).
2
(17.) Given /, + = 2,
1 + xxf/--
pht 7— = >^, a- then = \|r, - =
^ + ^^ i a. xi, ^ '^ 1 ^ I 1*
-; + "r- 1 + x>/^^
and the equation becomes
X
whose solution may be found by the method just explained
to be
X
hence
, 1
1 + X — 0-
X
(14)
(18). Required the equation of that class of cunres
which possess the following property, (Part IV. Fig. 1.) a
given abscissa A B =:a being taken, then the product of any
two ordinate^ at equal distances from B, shall always be
equal to the souare of the abscissa a. If y = yj^x repre^^t
the equation oi the ^urve, th^n the condition expressed ^nsUy-
tically is
\l/(a — x) ^y^ia + x) =? a*.
Putting a ^ X for j?, and then log \^ (ar) =: y\r^ t, we have
>//'j a: -I- x//-! (2 tf — a) = 2 log Oy
whose solution is
y\/^X =i log J -f ^ X — ^ (2 a — a:),
hence
log \/r J? = log a 4- »r -r- (2 a — x),
and
log a ♦.! --♦(2a--.r) ^ g,^'
yfrSC = S X S X • =
f(2a-«)
and changing the arbitrary function <f> into log 0,
ylr jc =: a ■!-• •
and the class of curves are comprehended in the equation
u =: L .
(2 a — x)
(]9). Given the equation
This is the equation on which the composition of forces is
made to depend in the Mecanique Coeleste, p. 5.
(15)
Put v/^j X for (y^r J?)* then it becomes
which 18 a particular case of
^1 j: + fl >/^,Q — !•) = 1 + V 1^ jr.
Substituting 5-1 for a, this gives
and eliminating x//^, fZ — j?) , we have
yj/^ X ss
1 -f a 1 - ^'^
niaking fl= 1 and v = 0, and changing ^ j: in 2 ^.t> we have
and therefore *
In case the coefficient of yjr ax in the equation
\^ x 4- /x >^- a a^ =/x, is of such a form that fx ./a * = 1 »
the denominator mil vanish, and we must then have recourse
to an artifice similar to that which has already been ex-
plained.
(20). Ex. 1. Let sl/'X ^ X^^y^rl := X".
X
put x//^ Jr 4- (j?** 4- V ^ j) \|r - = T**
X
• (16)
which coincides with the given equation if vzzO ; then chang-
ing X into ^9 we have
>^
X \ X/
and by elimination
yj^ X zz
1— (T*» + t?0^)rx-** + v^l^
a:*«d)- 4- x" •"^a? -f zJ^x.^-
and when v vanishes,
^x= f:l±f. , . (a).
• ^•"rf)- + x-*"0x
X
The equation in this example may be solved differently,
as follows. Multiply by x~" and it becomes
X — ">i!^x + x~\/ri =: 1,
X
put >i!^j X =: X "•" \/r X, then
whose general solution found by a process already explained
• - ■ » ■
IS
X^^X = - - 0X -h ^-. ,
2 X
hence
x** 1
y]^ X zz .^ — X*' ij) X -^ 2^ (p -
2 X
(17)
Thi$ solution differi in form from that which was pre-
viously foundj but it may be proved to be the same by the
following substitution ; since is quite an arbitrary function
this gives a:** 0- + a:""*" <^ a? =: 2 and (^i) becomes
X
X* 1
2 X
exactly as the last solution.
(21). Given
t
put y^r^x =s, JL ^ then it becomes
>/rx — 1
\x — 1^ a: — 1
V'
which is a particular case of
>^.« + (« + *0*)f . (-1-) = ^"^ ~ ^^\~ ^
with which it agrees, when a ^— \ and t> = 0.
Put- for Xj then
a? — 1
%/.» .+ la -^Vif^^^ .)>;.. J =-S__jL ,
whence by elimination,
t ®
(18)
(x — !)•— 1 1 + a + vd>x
1 7 X \*
1 — (fl + v x) f a + V ^ ^ I
and when a =:: ^ }, and v = 0, this becomes
and restoring the value of >//•, x, we find
!x*— 2x 0x _^
i
n
+ 1
^* = -T- : TT
>//- J? r: ^ ^ ^-2- :
x*«^ ^(1 -x') + (I — x*)^x
this solution was found by pui'suing the course so frequently
pointed out : another but not a more general one may be
obtained as follows : multiply by \/{\ — x*) 5 then
V(l -x*)x^x + x>/r ^/(l — x«) = xV(l-«*)f
putting >/(l — x') >^ X = >/r jX, we have
^t^ + ^. V(l - a:) = X v^(l - x»),
whose solution is >^, x= ^^ "" — ^ + 0x— 0\/(l — x*).
hence
^ 2 v^(i - X*)
(19)
It would not be difficult to shew the identity of these
two apparently different solutions.
(23). Given the equation
put >/^i^ = -7FT7r — ri> *^" >f^,(i-x)= yr^ f %
\/i^ (1 - x)i v/^y x)
and the equation become^
>^i^+>^,(l--a?)= 1,
whose general solution is
hence
yl^^x =
0*
•
(px
+
0(1.
-X)'
(>/.*)•
y,
(*
xY
putting 1 — X for x, and eliminating \/^ (1 — .r), we find
[0(1 -X) + 0;r)*
(24). Given (x^ar)« + ("^^-Y = il±:f!>;.:c . >^£-,
V X ^ x^ X
fl*
divide by \^ jt . >//- — then
a:
a'
y^x ^ _ ^ + ^* .
X
X
putting yj/^x =s , , this becomes
or
(«0)
, , , a' ar* + a*
a particular solution of which is >//-, j?s=x*;'hence
-21—- = A* or >^ a? = «* V^"^
^?
and the general solution of this is
1 + jr>^i
(25). Given Z±±JL^ x . f = 1+ »%
X
put V^i X s Jl , then the equation becomes
X
>/^, ^ + a:* x/^, - =: 1 + x%
whose solution is >^i x = yLZ — i^ ;
hence ^ = -^ ;
^l/'-r ibx -H ar*0-.
putting - for x and eliminating >/^ - , we find
•*" *. x^
(31)
(26). Given (yf. xT . (>/- - P^T - (^^T .(yl^-xT^^ ^5
putting ylr,xzz(yl^ x)*" . (x^ — xf, it becomes
whose solution is >^i « = r + x (a:, — •»),
hence
and by the process for eliminating ^(— x), we shall find
m
yj^xss
(27). Given yj^x +/^. >/r aar =/,x, where aa? is such
a function of x that a* i^ s= ,v ; putting a x for ;r^ we have
yj/^ax 4-ya X ."^X ssf^ a x,
and by eliminating \l^ax
If /a? ./aa: ss 1 and / x —/a? .f^ax =z 0, then the $(dtttion
becomes a vanishing fraction; also the general value of f^x
is in that case fxX ts \/{fx) .f^ {x, a x) and the equation
becomes
x//" 0? + fx .yj^aX zz x/fx ./^ (x, a x) ;
dividing this by \/(fx) and putting instead of ■ its value
\/(/ax) derived from the equation /*;r .faxzz'iy we have
which is a symmetrical equation^ whose general solution is
<p X -j- fpCLX
(28). Given a -^ b^x = ^(a + bx),
(29). Given -^ = + (^L£_) ;
o •\' c^ X \b + ex/
^x =
a*x
a« -_ ^*
a — ^
(30). Given -Jl£_ = + (-^) j
i)/«r s=
+
^
»-(-^% + (_i).'
2
where n is arbitrary; it may therefore be changed into any
synunetrical function of x and
X ^l
(31). The three last examples are particular cases ojF the
equation
a^^fX = 4/ a X^
whose general solution is 4"'' = <'"^-
(32). Given , ^ / x \ ° ^^ CtTtJ '
\ -^ nx
(«3)
(S3). Given a >}/ a t = 4^ «' •*•
4, J = a" a?.
(34). Given +r + a + (— I" ) = J 5
-— — is a periodic function of the third order, or of the
form o? x^x\ putting for x^ we have
and in this again putting for Xy we find
\ X ^ X — \
4/ ( ) and 4^ ( — Z — J being eliminated between these
three equations^ we have
• 4.x = — i— {i-fl(l -*) +-a'— ^1
l+«'tjr "^ ^ x-\S'
(35). Given 4^ x - « 4. ^^i^^lzi}^ = x^,
put ^— i 1 for jr, it becomes
yJL
Again, put ^^ for jr, and we have
«••*
*(7(i^))"'^'^ = (Ti?r'
(24)
by eliminating 4, f ^^^ "" — iland 4/ (-- -) from
these three equations, we shall find
(S6). Given ^. + ^ (±±Q +^Q^)=a
the function — is periodic of the third order^ and by
the process of elimination
4/ or =
*'^*(,^.)-K^)
• \
(S7). Given + — _ + ■- 2s a.
^ vAip v/+n^
Putting 4^,^ = — =, we have
V +x
*' ^ "*" *«^-^ + +1 ^ = ^*
a: 1 — X
whose solution is
1
a j: + 01 a: — ^1
(hX+^'^ + 9-
I - X X
hence
+^ = -J J— >■
(35)
^ 1 — X J
Putting >/^, a: = log yj/^ X, we find
1 X — I
^^ X + >/', -*-*-— + >^, 5fi log (tf'),
J — a? X
whose solution is found in the la^ problem; tlhangiiig 0i
into log <pt, we have *
^j?s=s ' ^' . log.""'
Similatly if a x be any periodic equation of the third order*
has for its solution
^ X + ^'dX + 4>f^x
(40). >/rJ?.>/^ax.x/ra«x = r*j '^
has for its solution
(4l)* Giten x^/^x 4./r. >|rax =/, Xj whfete a'xscx,
Putting successively a x and o' x for x> we hare
>//• a X +ya X . x^' o* X = /i a X,
-^ o* X 4-ya* X x^ X sis/, a* Xj
and elitnitiating \^ox and^ \//^a*x from thesfe thtefe eqi^tiohs^
we have
1 + fx.fcLX ,fa^X "
t D
(26)
(42). Given yj/^x + fx .yfrax =yi X where a* x ss-x,
a similar process of elimination will produce
> J, _ /iX-fx.f,aX-¥....fx.faX.fa''-^X.f,a'-*X
(43). Given the equation
1 J^ fx . (v^x + A^aj:) — \//^x.>/^oX = 0,
where a j? is a periodic function of the second order, and/«
is any function symmetrical relative to x and ax
/y\rX ,\lr aX — 1
X ^ f
yjrX + yl^ax
consider >//* x and >//* a x as two variables, and difierentiate
with respect to them, then
dyj^x . d^ax
+ TT"! — 4 . vi = ^9
1 +(>/.Jf)* (I + >^«x)*
and by integration,
tan-"*\^j?-|-tan""'>//'flx=C=f— rc->
whose complete solution is
11 ^ X , "^ 1
tan — * V j: =5 ^ tan""* — — . ,
0X + (pax fx
hence
>/^ a? == tan < l. tan""* II- ? >
C j: -I- a j: /x j
this process is analogous to one employed by M. Laplace, for
the integration of a similar equation of differences.
* Journal de I'Ecolc Pol^lecni^jue, Cuh. t5.
(27)
(44). Given
being a periodic function of the 4"" Order.
X—l V 1— X X X -^ l^
x.(l ^x)^l * 1 -x' X ^ x + IV
(45). Given >/.(x, ^) + >/. C^\ i'^ = 1,
Vx yy
x^(x, y).=x -.: — <p<^.yy
(♦6). >Kj, y) + V' (j, - 2/^= y%
«(*>!/) + *(-, -3^)
(47). Given ^ (^, y) + x» V C^! , -JL. ^ = -£1^ ',
Vx y— 1/ y-1
^(ar, y) + 0Cjr, -^,)
\ . y — Jy'
(48). Given ^ (x, y) + /(x, y) yj^{ax, /3y) e
^^/(J^, J') -/iC^j ° *> .y, ^ y),
(28)
where a* a' = t, ^y s= y and f(Xy y) is such a functiipQ that
1 1
then >/. (X, J,) -y(^, y)^(„,, ^y) +/(«*, ^Hx, y)* .
(49). Given ^r (x, y) = >^ (t:=^, IZJ)
(50). Given ^ (*, y) = ^t' C..£^2iJ(} , fJ^^X ,
> y . X / ^
(51). Given >Kx,^) «>/.(! y^|L, v^ 2^),
(52). Given y\r {x^ y) = x/. (y, ar)
(53). Given yj. {x, y) = (£)' >/. (y, a:)
V'Ca-. I/) « J .«(i, y) =-,.0. (X + y, xy).
(54). Given ^ (^ --. ^) « i^,
ax
differentiating ^K^r - x) = -r^j
m
putting w^x for a: in the given equation
and eliminating .-J—^^JHJ^ ^ we have
dx'
whence by integration
^x m b COS J7 + tf sin. T, ,
and it will be found that c ss — ^ ; hence
yj^x =s b (cos X — sin X),
(55). piyen >/.(*, y) = i^^l%A:il) .
put a -» y for j/if then
ax
diilercntiate this relative to x, then
dyjj^jx, a- y) _ d*yl^(x, y)
dx dx* *
which being substituted in the given equation produce^
whose spltttion is
>K^f 2/)« ''^y + «""0ii^j
and ^j being two arbitrary functions so constituted as t6
fulfil the given equation^ in order to determine theni| put
a^y for y and differentiate relative to x^ then
(30)
4>y = 4>ia -y) and <p,y = - 0, (« - ^),
whose solutions are
0^= X^yp « — y) and ipiy =3(fl-2j/)xi(^, a - y),
hence the general solution of the equation is
A similar mode of solution is applicable to the three follow-
ing equations.
(56> Given yj. (:r, y) = A >^(x, l\
ax \ y/
>^(x,5^)=.'^(y,-) +.-'_i.^.(y,-).
d
(57). Given \/' (x, 3/) = — x/^ (^x, — 2— ")
(38). Given >/^ (x, y) = -— >/r (x, ay), where.a' » = y
■^(x, y) = ,'^(y, ay) + «-'(oy -y)^.(y, ^).
(59). Given >/. (X, J) = ^il^i-^
where a is such a function that a*y ssy.
Substituting successively. a t^, a'y, a'j/ for y, we have
(31)
ax
From the given equation j ' ^^ may be eliminated.
by means of the second* and from the result ^ , ^ ^
ax
may be eliminated by the tlxird equation^ and continuing thisy
we should find
the solution of this equation is
^K y) = «'^y + •"~'0,y+ sin x.ip^y + COST. ^5 y,
0> 0i> 039 03 must b^ determined so as to satisfy the given
equation, taking the differential and putting a y for y, we
have
vW «y) sz^<i>ay'^t—'<p^ay+cosx\<l>^ay'-9inx.<l>^ay
a X
the first condition to satisfy is
0y=»0«y»
which gives
02/ = x(y* «y> <**y> *'y)>
the next condition is
0iy = - 0i«y
whose solution is
01^ = («'y - «'y + « y - 2/ (x. ^j «j/j «*y> «'i^)>
tlic other two conditions are
0a.y = - 03 « J/> i^nd 032/ =^ 0a « y
(32)
putting ay for 2/ in the second of these it becomes (p^oy
0s a^ y and this substituted in the first gives,
whose solution is 0, y = (a* y — y) x« (y> ° V* «* y> **'y)
hence
— -^ _. -1
and the general solution of the equatioii h
>^ (a:, 2/) = •' X (y* ''y* "^y^ *'^) +
+ t-'(a^y — o*y + ay — y) X. (y* ^f ^3 «*y) +
+ (9^y — y) ^« (^j «y* tt* J/* «'y) sin ^ +
+ («'!/- a y) x.(«y* ^*yr a*J^f y) iJoij?.
* • •
(60). Given the eijuation >/,(*, j,) = 0:^£l^,
where a is such a function that a'^x := x.
This equation may be reduced to the solution of the partial
differential equation
and the arbitrate functions of y which occur in its solution,
must be determined by the conditions of the eq/uation.
' (61). Given the equation
dyl/{a " Xj y) _ dyl^jx^ b -^ y)
dy dx '
put a — T for X, also i — y for y, then we have the tveo
equations
(33)
dy d X
_ d\l^{a-Xy b'-y) ^ dyf^jx, y)
dy dx
tf the first of these be differentiated relative to y^ and the
second relative to x ; then the right side of the first resulting
equation will be identical with the left side of the second,
and we shall have
d'^jx, y) ^ d^^{x,y) ^
dy" dx" '
the solution of this partial differential equation is
^(•^> y)^<P(^ -^ y) +<Pti^ - y);
the two arbitrary functions must be determined so as to
satisfy the equation ; we have
d'i^{a — Xj y) .If .V . / / \
dy
"^^^^^/^■^^ - ^'(6 +^- 2/) + ^'.(- ^ + ^ + y),
0' and 0', being the differential coefficients of ^ and ^, these
two expressions must be identical, hence
^' (/J - X - I/) = *' (^ + -^ - 2/)>
and
- 0', (/I - J^ + 1/) = 0'. (- ^ + X -f 2/),
the solutions of which equations are
and
and substituting these values, we have
t E
I
(34)
^(^, y) =Adx ^dy)x\ x+y, a-b- x-y\ +
+/{dx—dy)(a-6-^x—2y)xi } x—y, a-6-2x+2y } .
(63). Given the equation
d^(x,^) J^Q, y)
dx d^
Put - for y and differentiate relative to Xy then
y
dx"^ dx dy
Again, put- for t, and differentiate relative to y, then
dx dy dy^
hence
d^ yjr {Xy y) _ x^ d'ylriXj y)
di^ y* dx^ '
the solution of this equation of partial differentials is
^{x> y)^oc^ yjj + <Px (j^y) :
to determine the form of <ji and 0„ we have
d^ir (x, I) *
'. ^ =0(^^) + xy<p'{xy) + -i>\(z) ,
dx y ^V'^
ay ^ y ^^!^^' ^.; \x^
(85)
In order that these two expressions may coincide, we must
have
(X 2/) + Xt, <!>' (X y) =: - -L- i>' (J-)
X y \x y/
The first of these multiplied by d (xy) may be put under the
form
whose integral is
acy^ixy) = ^ (— ) ,
the solution of which functional equation is
s/xy V xyy
the solution of the second equation is
*-(p=\/i-^4-t>
employing these values of (p and ^^^ we have
(63). Given 1±^L^ = ^li^fjj^ ,
ax d^
where a*^ = ^ and /3* .r = :r a process nearly similar to that
1
(36)
by which the two last equations were solved will lead to the
partial diiFerential equation
day rf* x//' (JT, ^ ) _ /^ 3 X d* y\f fx, 1/)
dy 'dJ^ dx 5/ *
(64). Given y}/^ a X = yj/^ yj/ x = \l^*x.
It is evident, that whatever be the form of a, this equation
can always be satisfied by assuming ylrxssax, hence the
solutions of the following equations^
>^( — j:) = \//'*j? \^x=s— X
(65). Given . >/^ (2a — x) = x/^^ x.
Put x/'JJ' = 0""*/0j:,
then ^^ X =z (p-^ftpip-'fiiix :sz ^p-'fipx,
and >/^'x zz (j}—^/'' <p ip—^fip X = 0""'/'^x,
and the equation becomes
This equation may be satisfied in the following manner : by
making / a periodic function of the second order, we have
y*v = V, and the equation becomes
or
0(2 a — x) = 0x, ■
(37)
This is satisfied by making any. symmetrical function of x
and 9ia -^ X. As an example take /v s — t;, also
then
and
^
(66). Given >/. /^i 1\ = >lr'
VI -f "^y
xl/'X" = ^""*y^J? where ^ and y are determined by the
equations
(l>x = X \ f ^ \ .zAd /* X = X.
(67). Given yj/^a x = yj^^x, where o* v = v
putting >ir X :=. <p''^/(p X, we have r
(t>'^'f4>aX = (p — 'f^ (pX
determine ^ from the condition ^ a:=0 a <r ; hence.
i^ j: = ^ { JT, ax, o'*~~ ' X } ,
and lety*be such a function that f*x=:x, then the equation
is satisfied.
(68), Given yj^'' ax = yj/^ x where q >p and a^x ss x,
the substitution ^"^^/(px instead of \//- will give.
and this is satisfied if ^ x = % (x, a j?, a"— » x)
and also f'^^v = v, for it then becomes ,
ifi^^fp^ax=i<t>'^^fp<t>Xy where 0x = 0ax.
(88)
If in a function of two yariabiesi as i/^ {x^ y\ we sub-
stitute the function itself instead of one of those quantities,
the result is denoted thus,
if the function itself is substituted simultaneously for x and
tff it is denoted thus
(69). Given y^r^ (x, y) = a^
By means of the substitution <f> ~" ^f(p x, for >/^ jt, we are
enabled to reduce functional equations of any order to those
of the firsts a substitution nearly resembling it, will be of
equal value for those which contain two pr piore variables,
by assuming
xKx, y) = <p-\fi<l>x, tpy)y
we have
= «-*/^(0^* 02/),
and substituting this value in the equation
Put 0""* J? for a^, and ^""* for j/, also taking the function ^
on both sides
/*'*(^i 2/) = 0«.
If therefore we are acquainted with a particular solution,
we find the general one ; let the function ^ - be tried, then
y
(39)
A-
y
hence A — (pa, and the solution is
\,py /
a variety of solutions may be found of different forms, such
as
where « and /3 are any two homogeneous functions of the
same degree.
(70). If \Kx,y) s= a X + iyt
then \l/^(x, y) = (a + by— '{ax + 6y),
(71). If ^{x, y) is any homogeneous function of *,
and y of the degree n,
then
(72). Given >/.''•• ^x, y) = s/y\r(Xy y),
^(>c,y)^./}' + r »(l).
(73). Given
^••' (*» J/) = H^' y) + TWF)'
(40)
>(. y) = ^^^y-^ + (£±j)^ll>.
(74). Given >/-^(j, ^) = ! T t £' ^h
1 + >Kx, y)
>K*, jf) = 7f>r .
y^(l) +*»«(^-^
(75). ^i.^'iXytf) = F^|,'(X,y),
provided a and /3 are homogen^pus with respect to x and y ;
the first of the » + 1 degree, the second of the »"', and also
at the same time a(l, 1) = /3(], 1).
(76). Given >^^(*, ^) = i7x/,(x, ^).
Another solution of the same equation is
*<""=^(i^')'
where a and ^ are two such functions, that when j? = j/, we
have also
(77). Given x^'"^' (t, ^) = >^ (x, 5^),
(41)
(rfe). GiVeh ,!,"■» (X, y) = {yl.{x, y) ]
m
(79.) tSSven yfr^ix, y) == } x/.^« (ir, y) ] '
<80). Given yf^(x, y)=-.
y
(81). Given xv^»'»(j:, i/) = i/ >^''» (r, j/),
put 0""*y(^x, 0y) for >/r.r, then it becomes
putting ^'- * A' for x, and "" * y instead of y, we have
0-»j:.0-7>'«(j^, 2/) =:0-'i/.0-'/*'»(j;, y);
if /*'• (x, ^) a±y, and /*'* (x, y) = x, this equation, becomes
identical ; but making /(x, y) =fl— J?— ^, these two equations
are verified ; consequently the general solution is
(82). Given v|.''« (x, ^) . >/.''« (x, y) « xy,
>/'(•»•, .y) = 0-* Ct--^t-) •
(83). Given xa/^*^»(x, y) = a\/^»'*(x, y),
\ X ^
t F
Various methods for the solution of Functional Equations
may be found in the following writings :
Speculationes Analytico Geometricse, N. Fuss, Mem. de
I'Acad. Imp. de St. Petersburg, Vol. IV. p. 225. 1811.
Memoirs of the Analytical Society, p. 96. 1813.
Observations on various points of Analysis, Phil. Trans.
J. F. W. Herschel.
Essay towards the Calculus of Functions, C. Babbage. 1815.
Ditto, Part II. p. 179. 1816.
Observations on the analogy which subsists between the
Calculus of Functions, and other branches of Analysis,
Phil. Trans. 1817. p. 197- C. Babbage.
Spence's Essays, 1819. Note by J. F. W. Herschel^ p. 151.
Annals of Philosophy, Nov. 1817. Mr. Homer.
Journal of the Royal Institution. C. Babbage.
Various methods for the solution of Functional Equations
may be found in the following writings :
et
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